When it comes to design - choosing a font can consume a lot of your time, and it should. A great font choice can make the design and a horrible font choice can break it. Designer or not we can all distinguish the ugliest of fonts when we see them. There is a time and place for each typeface and there should be reason behind all font choices you make. For example you aren't going to pick Comic Sans for writing an email - it wouldn't be professional. Side note: You should probably never use Comic Sans. The truth is choosing a font is one of the toughest jobs amateur or not when it comes to designing.
There are many sites out there for downloading free fonts - the problem is most of them kinda suck. They're either too cheesy and can only be used for one or two occasions or they look exactly like Arial and Times New Romans, which isn't too original. If you plan on designing more, building a collection of great fonts will be really valuable to you. So lets talk about where to find the good stuff that will help you build your palette of fonts:
MyFonts is one of my favorites and I always check it first to see if there are any free gems that I can use for graphic projects. Make sure to type in "free fonts" when searching if you are trying to avoid fees, pretty free fonts do exist you just have to look.
There are some really great fonts on here that aren't offered in other places - this blog does the searching for you and narrows it down to the best fonts out there. I encourage you to check this one out - especially if you are looking for calligraphy or script fonts.
Deviant Art is good for all things graphics but they have a great collections of fonts that I love to pull from. I've found some great gems on here, I suggest Deviant Art for anyone who is just starting out and wants to build their palette of fonts - there are lots of zip files where you can download hundreds of fonts at a time.