Larry Bleiberg has written an article about places to go to get started in finding your family tree. Writing, in USA TODAY, Bleiberg suggests all sorts of things from the Family History Library in Salt Lake City to cruise ships and even at a five-star resort in Ireland (useful for Irish ancestry, obviously).
You can read 10 Great Places to Trace Family Roots by Larry Bleiberg at http://goo.gl/DhEfdJ.
As I read the article, I had to agree that each place listed was a great resource but I believe Bleiberg overlooked the best place of all: start at home. Talk to your older relatives first as they often can provide more information than any (expensive) trip to a distant archive.
If you still live within driving distance of your most recent ancestors, you also should check out local resources. Local libraries, courthouses, and other resources often provide much more information than do distant archives.
Visit a local Family History Center near you. There are thousands of them around the world and they provide huge resources. You can find your nearest Family History Center by starting at https://familysearch.org/locations/centerlocator.
Traveling to a distant archive without preparation and expecting to find information there is usually disappointing. You cannot walk in and say "Please show me the book of all my ancestors." It simply doesn't work that way. More than one unprepared would-be family tree seeker has been disappointed after visiting a major genealogy archive.
Always do your homework first. Learn as much as possible BEFORE you make the trip. The better prepared you are, the higher the odds of success.
As good as Larry Bleiberg's article may be, I would suggest it should be the SECOND article you read. I'd suggest the first one should be Getting Started with Your Family History at https://familysearch.org/ask/gettingStarted.