I guess I'll be the one person to recommend Adobe Premiere Pro.
Adobe charges too much and leaves lots of issues unaddressed, but if you're on this forum you're obviously already very accustomed to not paying for things, and Adobe certainly deserves that approach.
Despite the issues, Premiere Pro has the most robust and consistent tools, is pretty easy to learn, has a wide range of plugins available to enhance it, and is corporate-standard so the skills you learn may be professionally viable (if you care about that sort of thing).
Failing in that, Final Cut Pro on Mac is pretty much the same tool, just stripped down a little.
I've never used the Premiere Rush app on mobile, but the name implies a comparable experience, so maybe it's good too if you're working there exclusively?
My scriptwriting approach depends on the kind of content I'm making, but generally, I have a basic outline of what I want said with some extra details for any very important points, then I just try to follow the outline and expand on it live.
I do this because writing a full script can stifle you a little when it comes time to record, and a good chunk of on-screen talent will get EXTREMELY mechanical if you give them a full script to read from.
But obviously, if you go in with nothing prepared, sometimes it can go just fine... And other times you get the dreaded dead air. Not as bad here as it is on radio, but it's still never a good thing.
As for what to write them in... Well, any word processing software will do, really. I use Word 2007 but there's no right answer as long as you can keep your script handy one way or another.
What kind of content are you trying to make? What audience are you hoping to reach with it?