Don's sage advice which I'll share with you, got me going in this area.
He suggested that I start with a DeArmond, which are pretty easy to find and very reasonable. Basically they're Korean built Guilds with American made pickups. The two models you should focus on are the X-155, which is a larger size Jazz box based on the X-175 (my avatar) and X-500 or the Starfilre Special which is a thinner semi-hollow body (search old posts for an explanation). These guitars are built exceptionally well and hold their value if you decide to sell to move up later on.
If you really want the feel of a jazz guitar, go with the X-155. It'll do that, blues, rockabilly and anything else you want to put it through. The Starfires Special is also good for jazz, but really shines in blues and more gutsy playing.
Of course, once you have an electric guitar, you kind of have to plug it in to actually hear the thing. There are as many flavors of amps as there are guitars. If you're staying in jazz and blues, search around for a used Roland Jazz Chorus, Fender Blues Junior or Fender Deluxe which has more versatility. I've settled on the Roland with the X-175, they're really soulmates. For starters, you could also shop around for a little practice amp. Again (and others PLEASE jump in here), the small solid state Roland cubes are a great place to start.
I never owned an electric until joining LTG, now if asked, I'd have to say that my X-175 is one of my absolute favorite guitars, and the Bluesbird is tons of fun.
You'll get a lot of answsers to your question; listen to them all then have a ball!
