Ableton live i an awesome peice of software to use, but depending on which version you pick it could be up to $750. You can always try demo's, but if I were you I would look into the the Into version if you want to pick up Ableton. The intro version doesn't give you much in the way of sounds, but it does contain most of the effects. The beauty of this is that you can have ableton for your platform of composing/live performance, and with the money you save you can buy plugins, get free plugins(I beleive the plugins that Logan uses are free), or you can buy sample packs which have recorded loops and sounds that you can use in your music(such drum samples to make cool drum kits). Plugins(aka VST's) are a great way to be able to get all the differen't kinds of sounds you want. They range in price from free to hundreds. A simple google search of "Best free VST's(or plugins)" or "Best (paid) VST's." Speaking as someone who owns the top version of Live 8, called suite(they range from intro, studio, and suite) I can say that I hardly ever use the instruments and sounds that came with it, all 40gb worth! I personally love the vst's that you can get from Native Instruments. I personally love the Massive synth and couldn't do without it. Also their Reaktor synth while difficult to pick up at first, is if not the most powerful, one of the most powerful synth's out there. The way you use it is to literally reverse engineer a synthesizer from scratch to create your sounds.
Other options include Reason, Cubase/Nuendo, FL studio, and reaper. I don't have any experience with Cubase/nuendo or Fl studio, but I know Nuendo is the version of Cubase that's like a DAW(Digital Audio Workstation) but it let's you edit with video. Reaper is totally free(free to evaluate) and is actually a pretty solid program, and would be very suitable to learn on. But for $100 for the Intro version of live I think would be well worth it because in my opinion it's the best platform for performing live(so many dj's use it it's rediculous), and because it basically let's you connect just about any other program to it as I'll explain next.
The other option I didn't explain was Reason. I'm personally a huge fan of Reason, but it will really depend on your taste and workflow if you like it or not. What's differen't about reason is when you use the program it literally looks like the digital equivalent of an instrument you would use and tweak in the studio. When you load up instruments, you literally connect digital wires in the back and turn knobs, push buttons, etc. to create your sound. With both programs installed if you open Ableton, and then Reason it the two programs will sync together and you can feed midi(musical information) into Reason, and Reason will send the audio back to Ableton. With the unique design of reason it offers choices that no other Daw has because you can literally wire it do something special and unique, the posiblities are endless. The last version, Reason 5 would do you just fine and I just saw it on amazon for $200 which is really cheap compared to other software. The only downside to Reason is other than using it like I described above, you can't load an VST's or any third party software into it.
But before you spend money on anything I would highly recomend you look up some reviews, and most importantly download a demo of each type of software and play around with it to see what type of workflow works best for you. Not all of these are the best for performing live, but if your only going to be composing then which workflow is best for you.
Oh ya and I forgot to mention Logic which is also really great software as well, it's only available for Mac's.
Hope this helps^_^