Hello, welcome to the wonderful world of reeds.
I'm assuming you've never played a sax before? In that case, I 100% recommend you rent first, and take some lessons, before you think about buying. This way, you'll have a chance to see what playing the sax is all about, and see if you have the patience for practicing it.
In that case, your first question should be what kind of sax do you want to play? Solosists generally play one of three sizes of saxophones, either the tenor (Charlie Parker), alto (John Coltrane), or soprano (Kenny G), though the baritone sax is not without its charm. That might be a bit expensive for a new player, though. Additionally, the C-Melody sax, which is between a tenor and alto in size and tone is seeing a revival in modern jazz, but since these are no longer manufactured, your only option is to buy vintage.
I own two altos myself, and am toying around with the idea of buying a sorpano. There are several mixes on the site where you can hear these saxes in action. Geoffrey Taucer and The Prophet of Mephisto's Final Fantasy 6 mix "Canvas of Dreams" features a soprano in the lead, while AmIEvil generally solos on a tenor. I can't remember any mixes specifically with an alto, but you can imagine that the sound is somewhere in between that of the tenor and soprano.
So, brand names and what-have-you. When buying or renting, brand names are a must. Recently there have been a lot of horns comming out of China with, well, not at all stellar qualities. I'd say, on a scale of 1 to 10, they are on average a 2. Or less. For learning on altos and tenors, most agree that the Yamaha 23 series are the best - I'll get to Sopranos in a second. The two models to look for are the YTS-23 and YAS-23, tenor and alto respectively, depending on which you decide to learn on. Another brand to look for is Yanagisawa. When this company first started producing saxes in the 70s, the quality was similar to saxes currently comming from China. They've come a long way in the past 30 years, and their new saxes, I've been told, are quite good. Just make sure you get your hands on a recently manufactured one. A common name in the beginner's sax market is Bundy. My first rental was a Bundy. I can't really say anything nice about them, but they do work, and it's fine to learn on until you get a horn of your own.
Ok, Soprano saxes. These saxes have their own little troubles. If you're playing in an ensemble, they are notoriously hard to tune, and keep in tune. If you are very sure the Soprano is for you, don't bother with the beginner's lines. Arguably, the two best soprano models are the Selmer Mark-VI soprano, and the Yamaha YSS-62 (The 62 series is Yamaha's Pro/Custom model). Expect to pay at least $3000 US each for these, new.
One question I researched recently was buying vintage saxaphones. One reason for the popularity of vintage horns, other than the "cool" factor, is the quality. It is generally agreed that saxes made years ago were of much higher quality, on account of them all being hand-made. Arguably, the best saxes ever made were the Selmer Mark VI series, which were made in baritone, tenor, alto, and soprano sizes up until the 70s, when they started producing the Mark VII (Actually, they still make the Mark VI soprano). Expect to pay $3000 US for all but the baritone models, and more for a baritone.
I'll let you in on a secret though. If you want a professional, vintage saxophone at a great price, there are two names to look out for. Buescher and Conn. The Buescher True-Tone models can generally be found for under $500 US, and are some of the best American saxes ever made. Just make sure you buy one of the True-Tones manufactures between 1926 and 1950. The Conn 8M (tenor) and 6M (alto), have a similar reputation, though they are genereally slightly more expensive. Vintage horns aren't for everyone, though. They all tend to have unique "features" such as a slightly awkward mechanical system, or some sort of distinctive or odd positioning. It would be easiest to learn on a modern sax before buying vintage, but I won't say that it's better. It really all depends on the player.
The first sax I played was a Bundy (which is actually made by Selmer, in the old Buescher factory. Go figure), and as I mentioned before, I can't say that it was a stellar sax. The fingering was nice, though, it had that going for it at least. I ended up renting it for nearly 2 years. The first sax I owned was a Yamaha YAS-23, and I don't even need to sing its praises. It can do that itself. For a beginner's sax, it plays just as easily as any other beginner's sax, and the sound quality gets close to some of the professional-level saxes. I picked it up used for about $800 US from my local music store, and continued to learn on it. Eventually I did some small gigs, and that's when I first started to feel self-conscious about playing a beginner's model sax at paid performances. Now honestly, nobody would be able to tell the difference live, and it's hard enough in a recording, but I had played my 23 for years, and I decided it was time for a pro model sax.
My main sax now is a 1926 Buescher True-Tone, which I had overhauled and repadded with kangeroo leather, and I absolutely love it.
There are some other things to conside when buying a sax, such as reeds and mouthpieces, but I won't get into that too much. Your sax should come with a mouthpiece, which will get you started. Which reeds to buy generally depend on the mouthpiece, but personally I like Rico Royal and Vandoren reeds. You'd want to start on a fairly soft reed, such as a 2.5. Reed hardness is measured on a scale of 1 to 5. I don't know anyone who plays anything softer than 2, or harder than 4.5, though. Mouthpieces depend on the music you play. If you take lessons, either use the one your sax came with, or go with the instructor's recommendations. If you're just going to try and learn by yourself (not recommended!), the Selmer C* is a very popular and versatile mouthpiece. If you are looking for more of a jazzy-sounding piece, I'd recommend the hard-rubber Vandoren 55, 75, or 95 piece.
Where to buy, you ask? There is only one real option for a new player, and that is a genuine brick-and-mortar retailer. You can go in, try out the used horns they have for sale, get advice, try mouthpieces, and in general have a good time. You'll come away with a lot more knowledge that just reading anything I can write for you. Try a few music stores though, and keep shopping around.
eBay is a possibility, but I would not recommend it. Though I bought my Buescher through eBay, it was a perfect example of why to not buy musical instruments on eBay. It turned out that there was some severe damage to the octave key mechanism, which I had to have replaced. In addition, the sax was in overall poorer shape than the picture and description made it seem, which was slightly disappointing, and in retrospect, completely expected. If you must go through eBay, check the descriptions for "recently overhauled." That should mean it is in playable condition. Not "overhauled, then stored for XX years" because that could be problematic as pads dry out over time. Additionally, long-term storage suggests a mechanical problem or fault with the sax.
That's all I can think of for now. Good luck in your quest!
Edit: I didn't really explain at all the difference between a pro model and a student model. Student models are made to be very easy to play, and as a result, the quality of the sound generally suffers. Pro saxes are the opposite. They are harder to play, but generally produce much better output, which I would say is more suitable for recording. Live, I don't think you will hear too much difference. The price difference is generally about $2000 US.