This is perhaps one of the most commonly asked questions among beginners wanting to take advantage of GNU/Linux and its DAW, recording, composing capabilities. There are thousands of Linux distros out there and quite a few tuned directly for audio capabilities. The answer is whatever suits you, and perhaps that is the underlying theme among GNU/Linux distros. I have tried a few over the past few years and have settled on two.
I use Ubuntu Studio on my laptop. I had to sneakernet it (via thumbdrive) to my computer, so I am relatively stubborn on removing it to try anything else. It installed with relatively few problems, although I had to learn how to use NDISwrapper in order to enable my wireless card, and my laptop has an extremely funky soundcard and I still do not have microphone capabilities. But I would have had those problems with almost any Linux distro. Ubuntu Studio is really easy for me to use, although I have been using Linux for several years now. I am by no means a Linux expert, but I may have a slight advantage. I can see why some people would think Ubuntu is the easiest Linux distro to pick up and use though. Ubuntu Studio came with all my favorite audio applications preinstalled and I will cover some of them in the near future. I also did not have to do any funky kernel reconfigurating, it was fine-tuned for low latency upon installation.
My desktop system has Fedora Core 3 with PlanetCCRMA packages. This isn't the hardest way to go, but it also wasn't the easiest. Before the Fedora community flames my comments with questions as to why I'm still using FC3 while FC11 is now available let me say this, it still works. I spent countless hours learning Linux with this system, custom configuring things to work the way I want them to, installing funky software that had to be built from source because RPMs for these programs are still not available. I honestly like the way this system works. That and it's installed on an old machine that doesn't have a lot of RAM or processor speed, I'm not interested in upgrading. With that said, Fedora itself is easy to use, PlanetCCRMA was not, but a quick look at PlanetCCRMA's website now tells me that it is easier nowadays. PlanetCCRMA is a separate repository of packages for Fedora that will fine tune your system for low latency and also has a plethora of available audio software ready to go. There is also a nice easy option in their repository to install all their software at once.
I have also used DyneBolic in the past. It's nice to have a Live CD/Live DVD to see if your computer will even recognize what Linux is, but personally I like to be able to save my work hassle-free. I don't know enough about Linux to be able to manually mount external drives every time I boot the computer up. I did have fun showing other people some of my capabilities in the middle of Circuit City though.
My best advice to the novice is to aim and fire. Pick something and see if it works. If you've never used Linux before then your fields of fire are pretty wide. Linux-sound.org has a nice small list availabe at http://linux-sound.org/distro.html
Don't get frustrated though. This is your learning stage, so take the time to learn. If you've spent 3 hours installing something and it still doesn't work you're probably either doing something wrong or it's just not going to work. Don't be afraid to ask questions, but don't be afraid to dig through the manuals either. If you go to the #fedora IRC chat and get the infamous "RTFM" then it may be time to take a second look at the manual. (Trust me, I've been there a few times)