3 day pass is a definite, although my girlfriend and I only got a 2 day pass and stayed in King's Wharf on the final day.
Hamilton is a nice town, but pretty much all shopping.
I would do some homework and find a beach to go to along with Horseshoe Bay, which is beautiful, but I have heard there are many other quieter beaches to visit as well.
Be sure to visit St George's. Nearby we also took the bus to Crystal Caves which was nice.
We didn't take any excursions other than a snorkel trip which was ok. We explored on our own, and didn't feel like we missed anything other than a few other beaches (we are all about the tanning).
I am taking my little ones on the August 27th cruise. I have to agree with the other posters about April, I read in many places that the water is still chilly and the seas can be a bit rough. I looked at pricing when booking mine and the last 2 weeks in August on the Spirit was very reasonable compared to the June/July sailings.
Disney Magic "07"
Disney Magic "06"
RCCL Legend of the Seas