Bear in mind that rear mech cage would need to be big enough to accommodate the range of gears you're looking for. Don't go and buy a cassette with a big gear without thinking about whether you may have to change your rear mech to accommodate it.
To be honest, I wouldn't worry too much. When I took up biking 4 years ago I was fat, very unfit and asthmatic - I was terrified of the smallest hills and could never imagine myself getting up the big or steep ones. I went for 11-28. Just keep tackling them, slowly as you have to, and you'll find your ability quickly improves. Now I'm still overweight but did a very hilly 44 miles yesterday without feeling knackered at the end of it. Most amazingly my breathing is way better than it ever was, no wheezing or gasping. I recommend cycling to all asthmatics. I'm doing London to Paris in June, over 3 days !
By the way I applaud your wish to be able to maintain your own bike. I like to support my LBS, but there's very little on the bike I couldn't do now when needed.