Actually, the Shure SM58's aren't 'condensers', they're 'dynamic' mics, which is why I was suggesting getting yourself a condenser mic in addition to what you already have.
The Shure SM58's are kinda like the workhorse of the studio. Great for just about everything because they can handle both vocals and high decibels from stuff like drums etc.. A dynamic mic works differently and is more sensitive, so you never use it on loud sounds... like don't mic a loud amp or kick drum or whatever.

Condenser mics used to cost a fortune but these days you can buy some amazing one's for under $200. I bought the Rode NT-1 off eBay for $100.
It was practically new in the box, with a shockmount and carrying case.
So I'm basically sold on Rode stuff.
You can move up to the NT-2 and gain a rolloff switch and switchable pickup patterns.
Some mics are kinda legendary, like the Neumann u87... although you'll spend thousands. There's a weird looking tube mic called Blue that I've heard great things about too.
You'll also probably need phantom power for the mic. Most mixers have this built in, but it's best to check.

As far as the Alesis compressor goes... it'll probably work great, but you need to have the mic first. With vocals, you only want to use enough compression to get a hotter signal to tape but not so much that it amplifies every other sound in the room. Adding a compressor to a Shure SM58 will probably sweeten up the sound a bit and even things out so you can hear subleties better, but why try to fix something with another piece of equipment when you can just get a better mic to begin with.