You are really asking two different questions. Do you want to be a DJ or do you want to perform electronic music live? Those are different things. If you want to play other peoples' music and sit up in the booth at the club and get people dancing, then you need a DJ setup (technically, that's not a "booth" as the booth is the place in the club where the DJ goes, you just need the equipment). If you want to play music that you write (or remix) in a live setting, then you want a live setup, often referred to as a PA (rather confusingly).
If you want to DJ
A DJ plays other people's music and will mix two (or more) tracks together at the same time. The basics of this setup are 2 music players and 1 mixer. A classic setup would be two record players and a mixer. However, you could replace record players with cd or mp3 players. There are many, many choices.
You can also now DJ with a laptop. You put all your music on your computer and then use some software to DJ. Popular programs include Traktor and Serato, although there are many others. You could DJ in this manner just using your mouse and keyboard, but there are also many dj interfaces for software (Best Buy has even started selling these). You can also use record players or CDs as a control mechanism. There are tons of options here. The advantage is less stuff to carry around and more control, but it's harder to jump on a friends decks and if your laptop crashes, so does the party.
The biggest difference between software and traditional approaches is that software will automatically (or with some work on your part) match the beats of what you're playing. So the tempo will be the same. Some say this takes the art out of DJing, because matching beats smoothly through a long mix is hard. I think those people are pissed that their job is less secure.
If you go the software route, get a program, collect some music and start to mess around.
If you go the CD or Record route, you need to learn how to match beats. A youtube search of "how to beat match" will return lots of info. Do this first. Practice a lot. It's hard.
In either case, find a community and ask some questions. There should be a local DJ community wherever you are. Meet those people. Find people online. Start reading DJ mag or something. The traditional teaching infrastructure that almost all other instruments have does not exist for DJs. The best thing to do is make some friends who are better than you and drink beer at their place.
If you want to play your own music
Then you need a rig to play on. This can really run the gamut. You can use samplers and keyboards, midi, live effects, software, ad infinitum. This can become a huge (and I mean huge black hole for your paycheck). You can spend a literal fortune on synths.
There is good news though, you don't need to start there. To start out with, all you need is Ableton Live. This will let you play and remix songs, live, on the fly. You can do all sorts of stuff with this program (including more traditional DJing). It's about $500, but there's a free 30 day trial. You can get started by using your mouse and keyboard, but if you get serious about it, google "Ableton Controler" and pick something up.
In terms of getting started, just get the software and start screwing around. Then find some Ableton forums and start talking to people, looking for blogs, magazines, and maybe make friends with a local producer or two.