Update (7/28/16) : Four years ago, I wrote this post in which I listed the paleontological sites and museums that I thought would make worthwhile trips for anyone with an interest in fossils. Looking back at it now, I realized that I had a few additions (the University of Florida's museum and three German sites associated with the Messel lagerstätte) to make to it and that, as it might be a useful resource for people looking to plan paleo-centric travel, I should provide a permanent link to it on my main page. I've made both changes and will try to update the list more frequently than I have been in the future.
A few places on the list were no-brainers: the big museums of the East Coast and Midwest, the grand collections of Europe, or places like Dinosaur National Monument and Petrified Forest National Park where you can see spectacular fossils in situ. I added a few other places because they do an outstanding job displaying the paleontology of a particular site (Rancho La Brea, Urweltmuseum Hauff) or region (University of Nebraska Museum, Staatliches Museum für Naturkunde). A few places made it onto the list because of their especially impressive combinations of fossils and reconstructions or paleoart (the dioramas of the Denver Museum of Nature & Science or the murals of the San Diego Natural History Museum, for example), and a few more made it on due to the historical importance of the fossils there (the first described dinosaur fossil in Oxford's Natural History Museum, for example).
Obviously, this list is incomplete. It includes only places I've visited, leaving entire continents neglected. I'd be very curious to hear what else readers of this blog think would belong on it. Bear in mind that this list is intended for members of the general public, NOT for career paleontologists; we all have our own distinct ideas of which museums or sites are particularly interesting, and many of these may hold little to no appeal for someone with a more casual interest in fossils. However, I think any of the places listed below would be worth a trip for anyone with even a slight predilection for paleontology:
Map of Sites
United States
Petrified Forest National Park, Arizona
Rancho La Brea (reviewed 12/2015)/Los Angeles County Museum, Los Angeles, California
San Diego Natural History Museum, California (reviewed 3/2010)
Denver Museum of Nature & Science, Colorado
Florissant Fossil Beds National Monument, Colorado (reviewed 12/2015)
Dinosaur State Park, Connecticut
Yale Peabody Museum of Natural History, New Haven, Connecticut
National Museum of Natural History, Washington, DC
Florida Museum of Natural History, Gainesville, Florida (reviewed 2/2014)
Field Museum, Chicago, Illinois
Museum of the Rockies, Bozeman, Montana
Ashfall Fossil Beds State Historical Park, Nebraska
University of Nebraska State Museum, Lincoln, Nebraska
American Museum of Natural History, New York
John Day Fossil Beds National Monument, Oregon
Carnegie Museum of Natural History, Pittsburgh, Pennsylvania
Dinosaur National Monument, Utah
Field Museum, Chicago, Illinois
Museum of the Rockies, Bozeman, Montana
Ashfall Fossil Beds State Historical Park, Nebraska
University of Nebraska State Museum, Lincoln, Nebraska
American Museum of Natural History, New York
John Day Fossil Beds National Monument, Oregon
Carnegie Museum of Natural History, Pittsburgh, Pennsylvania
Dinosaur National Monument, Utah
Canada
Royal Tyrell Museum, Drumheller, Alberta
Canadian Museum of Nature, Ottawa, Ontario
Germany
Humboldt Museum für Naturkunde, Berlin
Canadian Museum of Nature, Ottawa, Ontario
Germany
Humboldt Museum für Naturkunde, Berlin
Grube Messel (reviewed 12/2015)/Hessisches Landesmuseum Darmstadt/Naturmuseum Senckenberg, Messel/Darmstadt/Frankfurt, Hesse
Paläontologische Sammlung, Tübingen, Baden-Württemberg
Staatliches Museum für Naturkunde, Stuttgart, Baden-Württemberg
Urweltmuseum Hauff, Holzmaden, Baden-Württemberg
Paläontologische Sammlung, Tübingen, Baden-Württemberg
Staatliches Museum für Naturkunde, Stuttgart, Baden-Württemberg
Urweltmuseum Hauff, Holzmaden, Baden-Württemberg
United Kingdom
Natural History Museum, London, United Kingdom
Oxford University Museum of Natural History, United Kingdom
Oxford University Museum of Natural History, United Kingdom