Most jazz guitar you hear on recordings is played on an electric guitar, so you'll want to get an amplifier along with an electric guitar.
If your goal is to emulate the same sounds you've heard other jazz guitarists use, which is that fat, clean, mellow and bass-y tone, then you'll be able to achieve that with a variety of guitars and amps, but the "classic" jazz guitars have always been the hollow bodied, (or semi-hollow bodied) guitars from Gibson, like the 175 (the ultimate jazz guitar) or the 335. Makers, like Epiphone, offer affordable copies of these guitars which are very nice in and of themselves.
For an amplifier, you have a lot of choices. If you wanted to walk into one of Greenwich Village's jazz clubs in New York City and play a gig, you would probably show up with your Gibson 175 guitar and a Polytone amp. I do not use a Polytone, but they are very popular with working jazz guitarists. They are compact and portable, I will grant you that. You could comfortably take your guitar and amp on the subway and show up to play a gig just about anywhere. But a Polytone amp is rather expensive and you don't need to spend that kind money on your first amplifier. Any decent combo amp (meaning with the speaker and amplifier built into the same box) that has a good clean channel will definitely suit your purposes. I would make sure that your amp has at least a 10" speaker. I have the very 1965 Fender Princeton Reverb (http://www.ampwares.com/ffg/princeton_reverb_bf.html ) that was used by jazz great Wes Montgomery himself and that amp only puts out around 15 watts into one 10" speaker but it sounds great! I don't know what your budget is, but if I were starting out today, I would probably get one of the smaller Fender combo amps.
You can play jazz on just about any guitar, the ones I mentioned, the Gibson 175 and the 335 are really the favorites and people are used to seeing jazzmen play these instruments, for whatever that is worth, but really, almost any guitar will do. You would look funny showing up at a jazz gig with some guitar that was designed with an aesthetic geared more for shredding and heavy metal, but really, there are so many other guitars that would be appropriate for jazz. Les Paul, who has a very well known guitar named after him, is a fine jazz guitarist and the Les Paul guitar is a solid body instrument. Lenny Breau, one of the most amazing guitarists ever, played a solid body. There are even some Fender Telecaster players who are so attached to that instrument who will insist on using them for jazz (The Telecaster crowd is fanatically attached to that guitar for everything. I should know, I'm one myself). If the sky's the limit, money is no object, and you really love the instrument as well as the jazz music you can play on it, maybe you'll commission a custom Eastman guitar to be built for you (one guy I play with has one, beautiful instrument), or seek out a classic D'Angelico or D'Aquisto hollow body jazz guitar.