Microphones and speakers are the PA system components that are MOST important for getting a good sound.  You may or may not have control of what speakers are used, depending on the performance situation.  But you can control what microphone is used.  For any vocalist it is a good idea to have your own microphone, because you never know what you'll run into when someone else is providing the PA system.

Recommending a microphone is a bit like recommending a good car to buy.  There are a plethora of choices, in a wide variety of price ranges, for a wide variety of situations (e.g., studio vocal recording, live vocal performance, instruments, guitar amps, drums, etc.)

There are two basic microphone types: condenser and dynamic.  An understanding the differences between these types is the first thing you need to understand in order to make a wise decision on what mic you should buy.

CONDENSER MICS:

They sound fantastic.  For this reason they are used extensively in recording studios.
They require “phantom power” from the mixer. (They will not work with a mixer that cannot provide phantom power)
They can be very large (for example, those giant mics used in recording studios are condenser types)
They can also be extremely small (for example, all of those tiny headset mics are condenser mics)
They are not generally used for live performance because they are prone to feedback
They are fragile (if one falls onto the floor you hold your breath... because it may be dead)
They are unable to handle extremely loud volume, so they are not good for drums or guitar amps
They are generally more expensive than dynamic mics

DYNAMIC MICS:

The best ones sound almost as good as condenser mics (but cheap ones sound horrible!)
They do not require phantom power, so they will work with any mixer
They are extensively used for live music performance situations
Some of them are very good at rejecting feedback
They are tough and reliable
They have more “proximity effect” (increasing bass response as you get closer)
The best ones can handle extremely loud volumes without distorting
They are generally less expensive than condenser mics

VOCAL MICS

Mics for live vocal performance come in both varieties.  Personally, I have some of both.  I love the sound of my old AKG C900 condenser mics (below). 

For quiet, low volume gigs, nothing sounds better.  They are crystal clear, and have fantastic detail and presence.  But they feed back like crazy at high volume events, and they are not rugged at all.

That's why for live performance, I recommend dynamic mics to my students.  They are much more rugged, and will hold up longer under the rigors of gigging.  Also, they will work with any PA system, they don't need phantom power, and the best ones are excellent at rejecting feedback.

For live vocal performance, the old "standard" dynamic mic that has been around since the mid 1960s is the Shure SM58 (below).  A lot of people still recommend the SM58 mic.  I don't.

It is true that the SM58 is the mic by which all others are judged and compared.  It is also the mic that nearly all pro sound techs use for live performance -- STILL... even after all these years!! -- for three reasons.  First, they are cheap (they sell for about $100).  Second, they are predictable.  Every sound engineer has used them hundreds of times and knows how they are going to sound.  Finally, they are almost indestructible.  You can literally pound nails into a board with these mics and they will still work.
 

Yes, the SM58 was by far the best mic when it first came out -- and yes, it’s still a decent mic -- but technology has improved in the last 50 years!  There have been better mics available for a long time, and today there are MUCH better mics available.  To be blunt, the SM58 is a dinosaur.  Anyone who still thinks the SM58 is the best mic out there either has no idea what they are talking about or has a tin ear.  Do a side by side comparison and you'll be amazed at the difference.  SM58's (and SM57's which are the exact same mic without the pop screen) are a complete waste of money.  DON'T BUY an SM58, and don't listen to people who tell you that you should.

My favorite dynamic vocal mic right now is the Sennheiser e935 (below).  It sounds almost as good as a condenser mic.  It is excellent at rejecting feedback and has a very high gain before feedback.  Sonically it has a very organic "presence" that sounds extremely natural.  I consider it to be the best all around vocal mic for live performance.  It is priced at $200 and I HIGHLY recommend it.  https://www.sweetwater.com/store/detail/E935--sennheiser-e-935-cardioid-dynamic-vocal-microphone

Sweetwater.com sometimes has a deal where you can purchase three of them for about $250 https://www.sweetwater.com/store/detail/E835S3Pack--sennheiser-e-835-s-live-vocal-microphone-with-on-off-switch-3-pack

WIRELESS VOCAL MICS

I understand the desire of lead singers who are not playing instruments to be able to hold their microphone and wander around the stage without the constraint of a cord.  With old style analog wireless systems, noise, hiss, distortion, bad frequency response, compression, interference, and other problems made using a cordless a VERY bad idea.  Modern digital technology has reduced these problems, but inexpensive cordless mics still sound horrible, because most of them use cheap analog-to-digital processors and lousy transducers (that's the microphone part of the microphone/transmitter combination).

If you spend many hundreds of dollars you can get a cordless mic that approaches the quality of the corded mics discussed above.  For example, Sennheiser sells an excellent wireless system that allows you to choose between different microphone capsules
https://www.sweetwater.com/store/detail/EWDHHSet-Q16--sennheiser-ew-d-skm-s-wireless-handheld-base-system-q1-q6

and you can get one that sounds just like an e935 corded mic https://www.sweetwater.com/store/detail/MMD935BK--sennheiser-mmd-935-1-bk-microphone-capsule-for-wireless-transmitter

It will sound nearly identical to a $200 e935 mic on a cord.  But that system costs about $800.  For that reason, I usually encourage students who don't have a big budget to use high quality corded mics.  They have the best sound, they are cheaper, and they are more reliable.

If you really have your heart set on a wireless mic, there is one system that I can recommend in the $400 price range (or $200 range if you already have a good mic.)  First buy an e935 mic for $200, then spend the remaining $200 on a separate transmitter and receiver for it. 

The best affordable transmitter/receiver I'm aware of is the Xvive Audio U3 (below).  I have not used it personally, but Sweetwater recommends it.  The specs sound good: it has 24 bit processing, excellent frequency response, low noise, low distortion, plenty of channels, and both the transmitter and receiver run off rechargeable batteries that charge with a USB cord, like a smartphone.  You just plug the transmitter into the end of your microphone instead of a cord, and plug the receiver into the snake or the mixer, instead of a cord.  This system sells for $200 at Sweetwater https://www.sweetwater.com/store/detail/U3Set--xvive-audio-u3-microphone-wireless-system.  (It has 5 stars on reviews.  By way of comparison, the Sennheiser XSW-D also has 5 stars, and costs $320.  And the Boss WL-30XLR sells for $250 and only has 4 stars.)

MICS FOR ACOUSTIC GUITARS

In the studio, never plug in your acoustic guitar.  Always use an excellent condenser microphone on it, and it will sound WAY better on the recording than it would plugged in.

But for live performance, it is the opposite.  Never mic an acoustic guitar at a gig!  It will sound lousy and the PA will start squealing with feedback.  Instead, save up and buy an "acoustic electric" guitar with built-in electronics.

If you have an acoustic electric guitar, there are two accessories you will need.  First, even with built in electronics, acoustic guitars can still cause feedback in the PA system, especially at high volume gigs, and in venues with hard surfaces that reflect sound.  If you get feedback using your acoustic electric guitar on stage, the easiest and cheapest way to eliminate feedback is to plug the sound hole with a d'Addario Screeching Halt Acoustic Guitar Feedback Suppressor (below).  It will ruin the unplugged acoustic sound of the guitar, but it will not have much effect on the amplified sound.  They only cost about $6 https://www.sweetwater.com/store/detail/ScreechHalt--daddario-planet-waves-screeching-halt-acoustic-guitar-feedback-eliminator

Second, you will need a quality guitar cord.  (LINK HERE and scroll down for information and recommendations for quality guitar cords and also about cordless guitar systems.)  Third, you should have a DI box (the above link also has information about what DI boxes are for, and a recommendation for a DI box).

MICS FOR ELECTRIC GUITARS

There are several brands of modern digital modeling guitar pedalboards that imitate tube guitar amps.  If you have one of these, you can plug directly into a PA system from the XLR balanced outputs ("DI"s), and you do not need a microphone for your electric guitar.  The audience will hear the electric guitar in the PA system, and you can hear it in the monitors.

On the opposite end of the spectrum, for players who are purists and want the very best electric guitar sounds, only a real tube amplifier will do.  If you have a tube amplifier and play in a band, you need to use a microphone in front of the cabinet to send a signal to the mixer.

My personal preference is halfway between these two alternatives.  I dislike the pedalboards because they just don't sound very good.  Also, part of the live performance experience is being able to hear my guitar separately from the monitors and to be able to tweak the volume and tone during the performance.  But I also don't love tube amps anymore, because I want a wider variety of sounds than a tube amp can give, and also I don't want the inherent problems of tube amps (inability to adjust volume without affecting tone, unreliability, and very heavy weight.)  That's why I use a digital modeling amp.  It sounds 90% as good as a tube amp, with half the weight, and all of the advantages of a pedalboard (large variety of sounds, reliability, etc).

Modern digital modeling guitar amps have balanced XLR direct outputs built right into the amp that enable you to plug a mic cord straight into the amp, just like a digital pedalboard does, so no microphone is needed.  However, in a side by side comparison, it almost always sounds better to stick a microphone in front of the amp rather than using the built in DI outputs.  That is because part of the sound shaping comes from the speaker and the cabinet.

If a professional sound company is doing the sound, they will have a mic that they can put on your amp, but you never know what you're going to get.  (Believe it or not, most of these guys are still using SM57's, which are dinosaurs.)  So it's nice to have your own mic.  You can use just about any dynamic mic (not condenser mic!), but a purpose built mic will have the best sound.  The best sounding guitar amp mic is the Sennheiser e906 (below) https://www.sweetwater.com/store/detail/e906--sennheiser-e-906-supercardioid-dynamic-instrument-microphone.  The Sennheiser e609 is a less expensive mic ($120) that looks nearly identical to the e906 and sounds pretty decent https://www.sweetwater.com/store/detail/E609Silver--sennheiser-e-609-silver-dynamic-supercardioid-guitar-microphone.

DRUMS

In large venues, drums always need to be mic'd.  If a professional sound company is running the sound, they will have a good set of drum mics for you to use.  But if you are playing with your own band PA system, you need to buy some drum mics.  Even in small venues, you need at least a kick mic and a snare mic.  In large venues, in a band with electric guitars, you need a full drum mic set.

Some drummers spend a bunch on their drums and cymbals and drum heads, and then buy a cheap set of mics.  They end up sounding to the audience like the trap set was bought at WalMart.  Don't go cheap on drum mics.  I haven't researched drum mics in recent years so I don't have a recommendation, but plan on spending at least $500 for a decent set of 4-5 mics, or preferably up to $1,000 or more for a high quality set of 7-8 mics.  How many mics do you need?  Well, kick drum, snare, each tom, hi hat, and overheads for cymbals requires 7-8 mics in most trap sets.  The kick mic is the most expensive.  A quality kick drum mic is $200+ just by itself.  The other mics are slightly less expensive, but the price does add up.  Buying the mics in a set instead of individually cuts the price almost in half.

MIC CORDS AND SNAKES

LINK HERE for a discussion of mic cords, and for my recommendations for quality, inexpensive cords and snakes.