I agree with everything Sarah says - watch out for dodgy dealers, although some are excellent. Maybe your instructor could recommend one, if she/he doesn't know of any suitable horses for sale.
Classified ads are fun, but you'd have to prepared to look at dozens before you found the right one - and it's amazing how the advertised '16.2 hh, 5 y.o. quiet to ride TBx' actually looks more like a 15.2 hh, 15 y.o mongrel and rides like a bucking bronco!
Avoid auctions like the plague unless you can go with someone very experienced. Apart from performance horse auctions, horse sales are often places where people get rid of problem horses (having said that, I bought my mare from an auction and she was a total bargain, but bargains are the exception rather than the rule)
Make sure you know the reason for sale and see if you can have a couple of weeks trial - you might get the horse home and find you're just not suited for some reason. For example, I used to have a big 16 hand thing with a trot so powerful that it made my back ache. It's not exactly why she was sold, but it was a factor.
Oh yes, and vetting! I didn't get my mare vetted and luckily got away with it - but she was an unbroken baby and wasn't likely to have any major physical problems that I hadn't spotted myself, or to be older than she was supposed to be. Plus, she was very cheap. Vetting might be a nuisance expense-wise, but if you don't bother you might end up with all sorts of horrific problems - seriously, a girl I knew bought what she thought was a four year old, which later turned out to be closer to 15 (the horse is still going strong in her thirties though, so it wasn't such a disaster!)
As for livery - I'd look for a place while looking for a horse and try to find somewhere asap. Nothing worse than wanting to buy a horse, but having nowhere to take it - and risking the owners selling to someone else while you find somewhere...
Finally, don't 'fall in love' with a horse and take it on, problems and all. Fatal mistake. I know, I've made it...not with my mare, but with another horse, who in spite of his beautiful looks, sweet nature and so on, is a complete devil to ride!
Best of luck,
Myrmex