Stephanie! Congrats on the new (to you) enlarger!!! What a perfectly serendipitous thing to have right now. You will have tons of time to practice until the wee one is born!
I can't speak for the chemical make-up of developers & fixers, but I do have some skin problems with my hands, so I use tongs. When I took a toning workshop, I also used disposable nitrile gloves (supplied by our thoughtful Gay Larson). There is no reason you cannot try both if it will ease your mind!
This is very exciting, and RC paper is a good way to go to get the feel of things. I agree with the folks who recommended that you buy a box of 8x10 and cut it down. Remember, that you will need to cut it down in the area you create as your darkroom... erm... while it's dark (safelight on).
I hope you will take Simon up on his offer! That book has tons of good info in it! Good luck & post up your practice prints!