Stitches can be used decoratively, or for their practical features.
You only need a conventional machine that can do zig-zag for the projects on this site, but if you can afford a serger and you expect to make a number of suits, I'd urge you to buy one: they're much, much easier.
If you have a normal sort of sewing machine, you'll have a choice of stitches to use - at the very least normal straight stitch and zig-zag. Ideally, your machine will have a range of stretch stitches in addition to zig-zag - overlock probably being the most useful.
The overlock on a conventional machine can be fiddly to use and doesn't stretch as well as a serged seam, but they wouldn't put on there without a reason.
For reasons I have yet to fathom, I prefer to use three-stage zig-zag rather than normal zig-zag.
Concerning zig-zag, something a commerical bodysuit pattern I've read recommended was sewing zig-zag without stretching the fabric particularly, then stretching the fabric to break as many stitches as possible, and the re-stitching, but stretching as you sew this time. I'm not really sure why they suggest this, but who am I to argue? :)
My understanding is that lycra is best sewn with a ballpoint needle. If you're going to sew through elastic you'll want to change to something sharper, or you'll end up poking the elastic strands out through the fabric beneath, which looks unsightly and tends to catch bodyhair (ouch).
A serger is a special - and usually relatively expensive - sewing machine. It can typically only do 'overlock' stitch, and it can only join two pieces of fabric along an edge so that they open out like a book, but the 'overlock' stitch it does is better than the overlock a conventional machine does - being stretchier and stronger because it is created with three or four threads rather than just two, and neater because a serger (overlock machine) can trim the cloth as it sews.
The stitch is a little complicated, but if you look at the diagram, the blue thread passes across the front of the cloth in loops, and threads through the tops of the loops of the red thread passing behind to keep the loose edges of the pieces of cloth together. These threads are held in place by the green thread, which is pushed through the cloth to loop around the red, and holds the blue while it passes to the next place where it's pushed through to hold the red. The four-thread variant of this stitch has an additional green thread, a little closer to the fabric edge. (The five-thread variant is just overlocking with a chain stitch running alongside the overlocking, but independent of it.)
I chose to buy a serger because I misunderstood something my friend told me, but I'm glad I misunderstood, because having bought it, it's so much easier to use for stretchy seams. If you anticipate doing more stretch-stitch seams than normal seams, and you can afford one, buy a serger as well as a normal machine. (You can just about insert zips with a serger, but it's fiddly. In principle, you can also create invisible hems with a serger, but I never got that working to my satisfaction - buy a normal machine.)