The Hawaii Tropical Botanical Garden, mentioned below is excellent. It has a lot of truly rare and interesting species, and the labeling was very good last time I was there. The East side of the island, the Hilo side is home to many nurseries, and many of them welcome visitors. There's an orchid nursery near the town of Volcano and the national park that is very good.
Be sure to visit the Panaewa Rain Forest Zoo, a few miles south of Hilo. It's plant collection is better than the animal collection. The local plant societies have planted their favorites extensively throughout the zoo. I planted quite a few palms there, myself, along with the rest of the local Palm Society. The rare plants are mostly labeled at the zoo. Admission to the zoo was free when I loved over there. Don't miss this place.
Check the list of sources of tropical plants in Hawaii. It lists nurseries. Note that a some of these are wholesale nurseries that don't want visitors. Just start calling them to ask if you can visit, but make it clear why you want to visit, so they don't get the idea that you're a potential customer if you're not going to be one.
http://www.plant-materials.nrcs.usda...ipmstn9017.pdf
There's a big garden in the middle of Hilo that locals call the "Tree Nursery." Mention that it's on Kilauea Avenue when you ask locals where it is. I think it's run by the USDA Forest Service Division of Forestry and Wildlife. The press release at this link about a tree give-away gives the address and phone number where it gives the Hilo location for the give-away:
http://hawaii.gov/dlnr/chair/pio/nr/2011/NR10-304.pdf
These aren't banana places, but they're fabulous places for enthusiasts of tropical plants. For banana places, try googling farms on the Big Island.