I used to live there, beautiful place. Definitely San Juan, If you want to see beautiful islands and cristal clear water you should take the ferry to Culebras or Vieques. You have to take the ferry from Fajardo (east side), very cheap and it takes about an hour to get to the Island. You can also go to Palomino Island, this is about 10 to 15 minutes and the water is cristal clear. The rain Forest is nice, is about 45 minutes from San Juan. You can travel the whole island in one day is 100mi X 50mi. If you drive around the island you'll find a lot of nice beaches. Like some one else said, just be careful where you go. San Juan is beautiful during the night, you'll love it, again just be careful. Stay were most people are and you'll be OK.
Definitely Old San Juan, El Morro (fort very similar to St Augustine), eat local food (is great), ask for Alcapurrias, Pasteles, there is a lot of great food to eat there. If you don't like any of that you'll find basically the same fast food that you would find here in the US.
I don't know much about fishing but I know there is a lot of fishing charters in San Juan.
BTW if you want to go to Palomino, Culebra or Vieques on a sail boat let me know I can contact you with someone I know in Fajardo. Here's his web site
http://sailpuertorico.com/, he is a very nice guy. I don't know about the price but his sail boat is beautiful. I use to live next to him.
If you need more info don't exitate to PM me.
Hope this helps.