I can do 10+ miles on my up-right cheap bike! Mine was cheap because it was used, and well, cheap. Early 80's Mixte style bike; and I got it because there are SO may styles of bikes out there, I wanted to know for sure what was right for me before I dropped some serious cash.
The 5 gears are a necessity for me. I think 21 is a little overkill for city riding, but I wouldn't say no to a 10 speed. You should look at mixtes, not as heavy and a little more practical than cruisers, but not as serious and aggressive as road bikes. My opinion is biased of course. Hybrids and comfort bikes would be good to try out too.
As for how long I've had the cheapo bike (and I say that lovingly)...not long. I'm already looking at upgrading - but just looking. I'm still trying to figure out what kind of rider I am, but commuting is a high priority. This is the one I'm thinking of getting in a year or so:
http://www.linusbike.com/models/mixte-8/
and who knows if I'll change my mind before then.
a lighter bike would be good - but I have heard aluminum frames really transfer every bump in the road to your body - you might want to read more about that if you have wrist problems (I have never tried it myself though).