yeah, composing is pretyt much writing notes after notes, or playing them on a keyboard and recording the MIDI signals as you press the keys (same result, it makes notes on the sequencer screen).
To produce trance, techno.. you will probably need a synthesizer (Novation SUPERNOVA, or a VIRUS B.. something like this) and also a sampler or a DRUM MODULE (MIDI device that produces only drum sounds). Hook everything audio-wise to a studio mixer, and hook all the MIDI to the computer. Get a MIDI controller (100 USD for a cheap keyboard) and hook it up to the computer as well. Install CUBASE VST 5 or LOGIC or CAKEWALK, this is your sequencer, where you will assign each track to a MIDI instrument, and start writing notes.
OR
Or you can get Fruity Loops (
http://www.fruityloops.com )or
REASON 1.0 (
http://www.propellerheads.se ) which are programs simulating a real studio, with samplers, syntheiszers, drumboxes... it's software, but if you have a good computer it will sound quite well.
As you are starting to compose, I suggest you to download the demos of these 2 programs, and start learning. Buy the one you prefer, and maybe in a few monmths or a year, you will invest more money and get some hardware.
As for playing live, this is maybe a bit early to say.. You will need to first compose all your music (home studio) and then you will be able to play live, but you still need to prepare it as a separate thing.
Good luck!