In Utah, the Family History Library is big and awesome. It has 2.4 million rolls of microfilmed genealogy records, 727,000 micofiche, 356,000 books, plus other periodicals and resources. This is where most genealogical records online come from. FamilySearch puts them online of course, but so does Ancestry and other sites. They have 200 special digital cameras for digitizing records, currently out in 45 countries. They have records from 110 countries around the world. Seriously, it's the all-you-can-eat-buffet of genealogy.
If you are lucky enough to be able to go to Utah to do genealogy for a couple days, this is definitely the place you will spend most of your time. They give tours, orientations, hold classes, and are staffed by genealogists and volunteers who are happy to help with any problems. I won't go into how to prepare for a trip there, because they already have a tips page online.
Now, if you are not so lucky to spend the day in Utah doing just genealogy, there are other options to access the info contained there. They loan out their microfilm for viewing at local family history library branches. Chances are there is one close to you (or at least closer than Utah). For a reasonably small fee you can order microfilm to be sent to your local family history center, and you get an e-mail when it arrives. Then you just go to the library and view it as much as you want for a couple weeks, and it gets sent back when you are through. It's pretty cool.
One thing to keep in mind though, is that not all microfilm is able to be loaned out. This can be frustrating. But, if there is a book or film that can't be sent to you, and you know the info you need is in there, you can do a photo-duplication request. This isn't for having someone else do the research for you, but if you have enough information for the form and just want a copy of the original record, this is a great service (and free!). For books and some films, they need a page number. If you don't have one, you can request copies of select pages of the index, then using that find the page number you need and send a new request for the exact record. How cool is that!
So, whether you can visit in person, have stuff delivered to you, or just use the online duplication service, the Family History Library in Utah is a great resource to check out.