I'm going to buy a Shure microphone SM58 type but I only have an old Peavey amplifier. The plugs are different. How should I do then to use my microphones, do I need a converter or to buy a new amplifier?
The Canon or XLR plug that's probably on the Shure mic tells it's a low impedance mic. If it has a jack plug, it'll be high impedance, and will work better with your guitar amp. You can buy adaptors which plug between mic lead and guitar amp that will match up the differing impedances, matching mic and amp. So, you need a matching impedance adaptor, female XLR to male jack.
A vocal microphone is actually 2 parts:
You can think of the preamp like a 1-channel mixer.
So if you are buying a brand new SM58 ($99) then also buy a small microphone preamp ($40) for it, like this one:
If that blows your budget, buy the mic preamp and then buy a $59 microphone instead of the SM58. That will give you better results for your $99 than an SM58 plugged directly into a guitar amp.
The mic preamp has a 1/4-inch output like a guitar, so that you can plug it into the guitar amp like you would a guitar, and the mic pre has an output gain knob that will enable you to feed a guitar-level signal into the amp, which is going to be just a little bit of gain, not the full line-level that the mic pre can give out. That will give you the best sound out of the amp as well as protecting the amp and microphone from accidental damage.
The mic pre will also enable you to plug your microphone into just about anything.
If you happen to also need a small mixer, you can get one of those instead of the mic pre and use the mic pres in the mixer.
If you want to connect a Shure dynamic mic (does not need phantom power) such as an SM58, SM57, Beta 58, PG58 etc. All you really need to make it work is the correct cable to plug into your 1/4 inch jack and mic. The cable you need has an XLR female plug on one end and a 1/4 inch TS plug on the other end. See picture below. Comes in various lengths.
Since your guitar amp is set up more for a high impedance signal and the SM-58 is a low impedance mic, you will probably get better results with a cable with a built in transformer to boost the signal. You can find an example here:XLR to 1/4 inch with Transformer
One other option is to use an adaptor which will convert the male end of a standard XLR mic cable to a 1/4 inch plug.
Or even better one like this with a built in Impedance Transformer.
Finally - the Shure PG-58 is offered with a choice of XLR to XLR cable or with the XLR to quarter inch cable. It's a lower quality higher impedance alternative to the legendary SM-58.
Any of these options will work okay as long as you keep your cable length less than 10 -15 feet. For long cable runs you may get some interference in your line without a balanced cable (like an XLR to XLR).
I once did that in a rehearsal session with a band I applied for. It was even an old bass amp. It did work and didn't seem to damage the amp, but don't expect anything that's close to a good sound. Speakers of Guitar or Bass amps are not build for such sounds, so it propably will be dull and bassy. Such an adapter for the cable you can get everywhere. My tip, just get an old, used PA box, maybe active, full range, they shouldn't be that expensive. I also recommend a pre-amp, so you can adjust the signal and maybe even monitor youryself.
"Shure" alone does not tell a lot. If it needs phantom power (like condenser mics do), it won't work on your amp.
Also you have to realize that a guitar amp is not general-purpose but intended to make a guitar sound good. In the line of instrument amps, keyboard amps are likely the best candidates for misappropriation by singers as they are more catered to reproduce the given sound rather than produce it.