Yes, I use my phone to record. The only area you'll run into issues with is if you're trying to get audio from it because phone mics generally aren't the best. You'd probably want a dedicated shotgun/lav mic.
I use Final Cut Pro. If you have a Mac, that would be my recommendation. If not, Premiere Pro is probably the best. iMovie is not bad for beginners but you'll soon discover that it's very limited.
And yes, an inexpensive tripod goes a long ways!