The downloading part I can't help you with, I find the idea of downloading movies kinda silly since it would take so long. Even with a broadband connection.
As far as recording them, you'll obviously need a DVD burner, a burning program, and some blank discs. The DVD burner we'll assume you have, as far as progs go, you can try Roxio's Easy CD creator. I can only half heartedly recommend it, because of my experience with it. A friend seems to use it everytime with no difficulties, while for me it's been trial & error. When burning a DVD you can create menus for it like a professional DVD, but the prog doesn't always seem to recognize a wide variety formats to burn. This could be a problem if you download a film, only to find out the program won't recognize it. I have version 7.5, only because I've read horror stories about the latest released version.
There is also Nero's, but I haven't tried it out. There's probably an effective freeware DVD writing program, but I haven't looked for them.
Your two DVD formats of choice are of course DVD - R & DVD + R. Supposedly DVD-R is the more common of the two, and should be recognized by more DVD players. Myself though, I've found that +R works better for me. You'll probably find you're going to have to try both formats, to be sure.
Yes, the DVDs will work on some DVD players. Some newer ones don't like created DVDs, while you'll find older and/or cheaper DVDs will more often play any DVD.
Hope some of this helps