I mentioned 5.1 because content is slowly but steadily increasing, plus there's concert dvd's in 5.1 that are coming out too. It's sort of hit or miss as to whether or not it adds anything but I figured I'd toss that in. If anything, you can start with 2 channel and expand in the future if you want.
As I was looking around at AVR's, it's just harder and harder to find a phono input on the less expensive units. I noticed that you wanted to use vinyl so you'll have to get a phono line converter/adapter/preamp or whatever else they are called. Here's an example of one:
http/www.parts-express.com/pe/showdetl.cfm?partnumber=245-876
It's $50, but one of the cheapest AVR I found with a phono input was $700 (aside from some really old model AVR's). Since I'm going to recommend a couple AVR's in the $250-$350 range, even with the extra $50, you will still come out ahead.
The AVR's to look at are the Denon's 1613 and E300, Pioneer's VSX-1022-k (Fry's has the cheapest one btw), Marantz NR1403 or NR1603. The Marantz are there incase you want a slim form factor but they are a little bit more. I'd suggest looking up the different AVR's and reading up on what they offer. From the sound of it, the Pioneer may be the best fit for you since it allows for multi room playback and it's $250 at Fry's. But read up on them and any questions you have come back and ask, more than happy to help explain things.
For speakers, in the budget we're working with I'd suggest going with the Pioneer SP-BS22 bookshelves. Normally they are $130/pr, but they go on sale a lot and currently Best Buy has them on sale for $89/pr. These are an excellent budget speaker and would be worth recommending even if they retail closer to $200/pr.
Glad to here you are ok with subs. A dedicated sub usually does a better job and handling bass than full range speakers. On top of that, the position of your speakers a lot of the time isn't the best place to play the low end, usually there's better spots elsewhere. Because of this, speakers now tend to be designed with the thought that a sub will be used. What you end up doing is having your AVR handle bass management which sends the bass that would normally go to the speakers to the sub instead under a certain crossover point. Just so you know, if you setup a second room, you won't have an output for a sub there so you'll want more full range speakers.
As for a sub recommendation, I'll mention the NXG NX-BAS500 which can be had for around $250. Because you specifically mentioned jazz, I'm staying away from cheaper subs which tend to be less accurate and boomy in certain note ranges, not ideal for jazz or blues for that matter. Here's a review from a person who deals with budget subs as a hobby:
http/www.hometheatershack.com/forums/speaker-subwoofer-reviews/67178-nxg-nx-bas-500-subwoofer-review.html
The problem right now is that it's out of stock and I haven't seen anything as to the status of this model.
As another suggestion the SVS SB-1000 which costs $500. This is a solid, accurate sub. It may seem like a lot of money for a sub but it's worth the money even in a budget setup. For accuracy and flat response with the capability to fill somewhat larger rooms, this is the sub that tends to be mentioned as the starting point on the low end of a decent setup. Given that the Pioneer AVR and speakers come to $350-$400, even with this sub you will be at $900, $100 below your max.
It's something to think about, but the NXG isn't a bad choice if you decide to go with that instead.
Any questions, thoughts, concerns, please ask.