I've been doing Let's Plays myself for over two years, and I've been using Open Broadcasting Software [https://obsproject.com/] the whole time. It can be a little finnicky to deal with, but it records mic and PC audio on two different volumes, can put in multiple visual feeds at once, and doesn't swallow up craploads of space. Is it the best? Probably not, but it works and it's free. I'd say that's good for me.
Editing software I can't help you as much with, as I've generally just used Windows Movie Maker.