Advice for a MP3 newbie

What you need is a "flash" player- that is, a player that uses flash memory (no moving parts), instead of a hard drive based unit.
I just got an iaudio U2, and I'm really really impressed with the quality of this little marvel. I want mine for the same reason you want one- to use when running, etc. One thing that I don't like about the U2 is that it doesn't come with an arm band. I ordered an armband for my U2 from the iaudio website, but if it doesn't *flawlessly* integrate with my player, as far as I'm concerned all the awesome features are for nothing if I can't wear the thing when I want to listen to it. The iaudio G4 (I think that's the model) is really nifty too. I think it comes with an armband. You probably won't find an AM radio yet- manufacturers typically leave out certain features so they can sell "improved" models to the same people later.
Check out the iaudio models. You can also do things like record from the radio and encode MP3s from external sources without a computer (copy your CDs from your CD walkman, for instance).

JK

The RCA Lyra 2012 might fit the bill.
It is not the tiniest player (though it is small), or the best looking. But it uses AA batteries to get 50 hours of continuous play, has a built in FM tuner with the ability to record radio, it can take external SD cards so you can upgrade it from 256mb to 1280mb later if you want.

If you're looking for something a little more attractive, and smaller, and are willing to pay a lot more (the RCA is under $100), then the iRiver IFP799 is a good way to go. It runs almost 40 hours on it's battery, has 1gb built in memory, is tiny and pretty nice looking.