1. Huntington Botanical Gardens - In Southern California, this private garden housed 14 themed gardens, as well as art galleries with regularly changing shows and the famous Huntington Library. Be sure to visit for high tea.
2. Butchart Gardens - A jewel of Vancouver Victoria, the gardens are a magnificent display of Victorian tradition tended for generations by the Butchart family. It has 55 acres that are open to the public.
3. Brooklyn Botanic Garden - With 13 themed gardens, as well as the conservatory and other collections, the Brooklyn Botanic Garden is an oasis that had humble beginnings as an ash dump in the 1800s. It celebrated its Centennial last year.
4. Denver Botanic Gardens - Encompassing 3 unique sites, the Denver Botanic Gardens were founded in 1951. It is one of the first gardens to emphasize native plants as well as water conservation and biological control of pests.
5. The United States Botanic Garden - Located in Washington DC, it is one of the oldest gardens in North America. The Conservatory allows a year-round escape into the beauty of the displays, Bartholdi Park showcases home landscape design and the National Garden displays the finest in Northeast native plants.
(Images: As linked above)
Before & After: Emily Henderson Transforms a Senior's Studio Space
Monica & Austin's Charming Bungalow
Good Food to Share 
The Big Book of Small, Cool Spaces 
Apartment Therapy Presents 
The Eight-Step Home Cure 
Ercol Bar Stool
How To Cook Fluffy, Tasty Quinoa
True Greek Recipe: Paximadi Bread Salad with Tomatoes, Feta and Capers
I'd add Brookgreen Gardens in South Carolina, south of Myrtle Beach
Great list!
I'd add Red Butte Garden, in Salt Lake City, UT
Butchart gardens is in Victoria, not Vancouver.
Brooklyn Botanic Garden is a true gem, and easily accessible from Manhattan by subway! The photo shows the Garden at peak Cherry Blossom bloom in early May. Check out the Sakura Matsuri Festival every spring for a Brooklyn version of the annual Japanese cherry blossom festival.
For variety, I'd recommend at least one Desert and one Tropical Garden.
= Desert Botanical Garden in Phoenix AZ http://www.dbg.org/ OR Karoo Desert National Botanical Garden in South Africa http://www.sanbi.org/index.php?Itemid=333
= Fairchild Tropical Botanic Garden, FL / National Tropical Botanical Garden, HI
@wiggles, I was confused as I was looking at their site. It says, "Vancouver Island, 23 km (14 miles) north of Victoria," so does that mean it's not really in either city? I visited when I was 16, so I don't actually recollect the specifics.
The Luther Burbank gardens in Santa Rosa and the LB experimental farm in Sebastopol are free and full of roses and Burbank hybrids.
http://lburbank.users.sonic.net/
When you're at Butchart Gardens, I highly recommend having high tea--lovely tasty treats (and you can have coffee rather than tea if you want to).
Wow, I just realized I've been to three of the five gardens listed. Kudos to my parents for taking me everywhere.