There were many different reasons, but I'll focus on three. The first, as you mentioned, was to experiment with new designs. The second was to experiment with weapon effects. How did different bombs affect different things? The third was political reasons.

Operation Crossroads was a series of two weapon tests (three were planned) that dealt with the latter two reasons. The Crossroads detonations, which took place in 1946, were the first since Hiroshima and Nagasaki and used plutonium implosion bombs similar to the ones used on Nagasaki.

The tests were proposed as a way to demonstrate the effects of atomic weapons against ships, and they were widely publicized as a way to show off the United States' latest weaponry. Extensive measurements were taken of the twin blasts, and test animals were used to measure the weapons' effectiveness.

Wikipedia's article on Operation Crossroads is featured content, and it's not a bad primer.

After Crossroads was the Sandstone test series, which included three detonations that tested new bomb designs including what would become the first mass-produced nuclear weapon.

Subsequent test series would test newer warheads and delivery vehicles, the effectiveness of atomic weapons against structures, the ability of soldiers to operate in post-detonation environments, and many other factors. The last confirmed belowground test (aboveground testing having stopped in 1980) took place in North Korea. Other nations have begun using supercomputers to simulate atomic detonations, thus replacing the need for physical testing.