We used the thick cotton training underwear. It helps contain the pee if they just start to go a little, but if they full-on pee their pants it's going to be dripping down their legs. We bought some of the vinyl "covers" but ditched those after 1 day because they were so uncomfortable and also because we found they do contain the mess enough that the kids would just sit in it and we didn't know they'd had an accident right away.

Pick up some "nature's miracle" - it's an enzyme-based cleaner you can buy at most pet stores that really does eliminate pee odor from carpets, furniture, etc. Even if you go with the cloth/vinyl undies you are bound to have some accidents on the carpet or furniture eventually.

We did potty training right after Christmas and just went cold turkey and put them in underpants. The first couple days were hell, but we saw a ton of progress by the end of the third day. We used pull-ups just for naps and overnight. We called them "sleeping underpants" and took them off first thing when the kids woke up.

Good luck!