Ben, Fedora is an excellent distro, but it is FOSS (Free & Open Source Software). That means that the codecs you'd use for much of the video watching and gaming would not be included in the initial download. Instead, you'd use one of the third-party repositories, like Livna, to add the codecs and set up your system for your needs. You should also realize that the primary purpose for Fedora is as a test-bed for new software and technologies, so it's rapidly changing (lots of updates), new versions are released about every six months and only the last two versions are supported.
Now, if you want to learn about linux, get your hands dirty and take some time to configure things, Fedora's the distro for you! If, however, you simply want a distro to immediately handle your needs, I'd recommend Mint Linux (a variant of Ubuntu), PCLinuxOS or Mandriva. Those will include the necessary codecs and/or be very simple to set up. Support is also longer for each of those.
Before deciding, I'd try a Live CD version of the distros and see which ones work well with your hardware and that you like the 'feel' of. Keep in mind that gaming is a mixed bag with linux. Most of the games you may be playing now are not necessarily supported in linux and may require workarounds ('wine' and 'cedega' for instance) to function or may not play at all. Many, many people dual-boot for that reason. On the other hand, there are many linux games that are also excellent and you may enjoy as much or more.
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Don't use any of my solutions on working computers or near small children.