It can easily be done on a regular machine, but it is much easier and the results better with a serger/overlock machine. I've made all of both my girls' leos and swimsuits for many years. Jalie is a really good pattern company in general, their gymnastics Leo patterns are decent and far better than anything else commercially available. For ballet leos I have ended up drafting my own as no commercial pattern fit the bill, but for gym leos.... Jalie is the best.
If you attempt to do this on a regular sewing machine, use a narrow zigzag for the joining seams (don't be tempted to use the "stretch stitch" if your machine has one.... It puts too much thread in the seam and doesn't work well). Don't use a wide zigzag unless you want funky looking seams. A narrow zigzag will allow the seam to stretch slightly but will look good from the outside. Don't use a straight stitch, it will pop.

For the elastic openings I would suggest zigzagging it on in the round, folding over and stitching it down using either a triple step zigzag or a twin needle. I use my cover stitcher, but lots of home sewists don't have one of those.
It's really not difficult, but I would suggest practicing with scraps first if you're very inexperienced. Some of the Leo fabrics can be slippery and it's much better to get the feel for it first rather than messing up with expensive fabric!
The fabric definitely has to be 4-way stretch, and for lots of fabrics you're best off using a 4-way stretch lining fabric in the front also (yes, even for little ones).
Regular polyester thread is fine, I have never had any problems with that.
Don't be discouraged if your first one isn't perfect, they rarely are.... I have been sewing for many, many years and I'm still learning and perfecting! I sew for a living and am very confident in my skills, but there is always room for improvement and my leos definitely look better now than they did years ago.

best of luck!