Like Fede said, FL-Studio is one option. It's pretty widely used, and recommended, especially among internet musicians, as well as beginners. Very capable DAW, though I'd recommend trying a couple out.
Me, personally, I fiddled with FL-Studio for a little over a year, and eventually came to the conclusion that it wasn't for me. For some people, it's everything they could want, but personally, I wasn't feeling it. Looking back, the reasons were that it felt too cluttered, especially after I'd get a lot of different instruments and VST's loaded up, and that everything was laid out so.. Hmm. Nontraditionally, I guess?
You have your more "professionally"/traditionally laid out DAWs, the ones you'll see in studios like Pro-tools, Cubase, Sonar, Logic (Favorite of Mac users) and Studio One to name a few, which are as great as any other DAW, depending on how you personally understand them. Then some that have a mix of traditional layout, with their own quirks, like Ableton, Bitwig (One of the only major DAWs with native Linux support), and Reason. Lastly, are the ones that are pretty odd, being.. FL, and something else, probably. FL is the only one I've ever looked at that's so different, but there are probably others.
OS support is probably important to you as well.
The only DAWs that have native support on Windows, Mac, and Linux devices, are Bitwig, and LMMS (An open-source, kinda watered down DAW), to my knowledge. Don't quote me on that, though.
Windows and Mac are Ableton, Pro-tools, Cubase, and jeez idunno what else.
Strictly Windows is FL Studio (At the moment, they're working on OSX support), Sonar, and maaaybe Studio One?
Mac-only is Garage Band, which comes free with your computer, and Logic.
There are probably more in each category, but that's what I know of.
My personal recommendation is Ableton. I find it cleaner and with less windows than FL, as well as a million other things, but preferences, you might completely disagree. (Edit: I can also actually link you to tutorials for understanding Ableton if you wanna try it out, can't remember what I used for FL back in the day. SeamlessR is good if you go with FL)
Lastly, what are you looking to compose? Electronic? Orchestral? Wanna plug in a guitar and track? Let us know, and I'm sure either Fede and I could recommend some libraries, VST's, or other things.