I can recommend Reason - maybe a bit of a steep learning curve if you've got no previous experience with sound design or music production, but, hey, I learned on it Only trouble is it can't do recording audio (unless you get Record as well), and it can't do external plugins, but I've not found that so much of a problem - if you get learning about synthesis, you can do pretty much anything with it. Some people prefer FL Studio over Reason, but I tried it for a while and could never get into it; you may want to try it out, but IMO it's awkward to use. As you progress, you might want to upgrade to a more serious DAW like Cubase, which you can still use both Reason and FL Studio with.
In my experience, LMMS is a bit too unstable for serious use (on Windows, less so on Linux but still not great), and I'm not a great fan of the interface. You might want to try out a tracker though, there're plenty of free ones about; they can be a little bit tricky to get to grips with at first, but they're pretty damn efficient once you get started. MilkyTracker's a good one to try out. There's also seq24, which is quite a nice, free, loop-based MIDI sequencer, though you'll need something to make a sound with as this only generates MIDI note data.
All in all, choosing your software comes down to how you work best - I'm used to hardware so Reason (which looks like a studio, of sorts) works great for me. You might prefer to track, or you might even want to stay entirely in hardware, and get something like a Korg Electribe. This is something you'll have to find out by yourself; get demos and play around with them.
General advice: learn everything you can about synthesis, sampling and MIDI. Wikipedia is your new best friend. Install MIDI-OX and MIDI Yoke.
Hope this helps
