Combee - you're getting some good advice here, and some that should be taken with a few grains of salt. I don't post that often on the BG side here, but here's my two cents on the matter...My opinion on the options given so far.
First, I don't see anything wrong with going for a quality instrument right off the bat. There are many pro's to starting with something cheaper, but many con's as well. I started on a cheap Hamer Slammer P-bass that was (literally) made of plywood, had no sustain (not in a cool, motown sorta vibe either) and was hard to play even after numerous setup tweaks. I had played guitar for many years before that, so I had some knowledge about how to set an instrument up, which helped... but when I graduated to a better quality Yamaha after about a year it was like night and day. Things that I was struggling to play before on the Hamer became accessible very quickly once my fingers didn't have to work so hard.
Yes, the initial cash outlay is less with a cheaper instrument - but another way to look at it is that the resale value on a Fender (if for whatever reason you wind up not pursuing bass) will be higher than something entry-level. Especially if you're buying the Fender used in the first place. If you buy a cheap instrument new, then sell it, you'll lose money. If you buy the used Fender, and then sell it - you'll probably be able to recoup what you paid, or close to it.
Having said that - I'll add a +1 for Ibanez. Check them out, I picked up an older SR205 last year that kills. Paid 200 bucks for it. Every bit as good as that Yamaha (my first "good" bass from many years ago) that I paid 5x the money for. The higher end models are even nicer - if you can afford an SR504 there's no reason that bass couldn't be with you for a long time.
Re: hand size - that's a funny thing sometimes. I took an old GF bass shopping once, she had tiny hands (I mean Tiny) but her forefinger-to-pinky stretch was a good inch more than mine. We were originally looking at short-scale models, but after we did that little test we realized she could handle a 34" scale no problem.
(She wound up buying a guitar instead, in case you're wondering. Should have been a sign

)
The thing to do is try out everything you can, different neck sized, scale lengths, and find something that feels comfortable - and then you have a starting point and know what to look for.
Okay, last point, on 1 vs 2 pickups.... In addition to the tone options (more bass from the neck pickup, more treble/mids from the bridge) two pickups can be used to even out frequency response/dynamics...like a form of natural compression. To me it's one more reason to have two. On the other hand, I really like the big, punchy sound of a P pickup - and some really good players get a wide range of tones out of that one pickup. My main bass for gigging right now is a Godin with a P/J setup... If I want a P sound, there it is. Or I can blend in the bridge pickup just a bit to even things out, and generally get all the other sounds you'd expect out of two pickups. Overall it's a bit darker sounding than a J when both pu's are wide open, but it's a cool option if you're on the fence about getting a Precision or something with more versatility.
Hope some of all that helps, Good Luck shopping!