I'm familiar with the Trinity test but were there any others?
They only had one full-fledged test — Trinity.
They did a lot of other experimental and diagnostic work short of full testing, though. This included:
Near-criticality experiments ("tickling the dragon's tail," as they called it), where they created an almost-critical or just-critical state of nuclear material that helped them figure out exactly what would probably happen when the bomb went off (this was especially helpful for the uranium bomb design, and one of the reasons they didn't feel they needed to test it to have an idea of how well it would work)
A large, conventional explosive experiment (the "100 ton test," named for how much TNT was in it) laced with radioactive material to get an idea of how to calibrate their instruments and the probable distribution of radioactive byproducts
Lots and lots of experiments designed to help them estimate the efficiency of the bombs without actually setting off any fissionable material (the "RaLa experiment" is the most famous of these)
And of course they tested the non-nuclear sub-components of the bomb hundreds if not thousands of times to get an idea of their reliability
So, yes, they only tested the full bomb once, but they were not going into all of it completely blind. The reason they tested the bomb at Trinity is because even with all of that testing, they couldn't accurately predict how well it would work. They knew the theoretical upper range of the explosive (over 100 kilotons), they knew the likely upper range of it in practice (around 50 kilotons), they had an idea of what the most probable output would be (4 kilotons), and they knew, of course, the lower-end of the output (around 2 tons of TNT, which is just the high explosives in the weapon). The simplicity of the uranium gun bomb's design meant they could estimate its yield very accurate (they thought it would be about 15 kilotons, and they were right), but they had bigger uncertainty with the plutonium implosion design. The test revealed that it worked better than they had expected (20 kilotons).
Just the one. The US had 3 bombs: 2 plutonium based (in which one was tested at tri ity), and one uranium. There was no guarantee the uranium one would work until they dropped it on Hiroshima.