Castillo San Cristobal & El Morro
These two forts, which protected Spanish-controlled Puerto Rico from both land and sea-based attacks, are both awesome to explore and interesting to learn about. We climbed up and down and all around at both sites, something we could never do if these forts were located in the States. The entrance fee is $5, good for both locations (as long as you visit each within a week of each other). I recommend exploring one, then taking free transit to the other.
Walking/Riding on the Streets of Viejo San Juan
Old San Juan is small enough that it's easy to take a self-guided walking tour. Another option is hopping on the free trolley that makes its rounds on the cobblestone streets. You can take in the sights and sounds of the city from the (relative) comfort of a bus.
I only have one actual restaurant recommendation, because the French toast at Caficultura was the bomb.
Good For: Feeding yourself before a long day of walking and exploring.
Ambiance: Friendly Brunch.
Notes: Get the French toast. Everything else was amazing too, but coconut French toast with pineapple marmalade? You don't see that every day.
For other meals, scope out the menus on restaurants as you walk by. Find something with mofongo or tostones, get yourself some seafood, and definitely treat yourself with ice cream from a street vendor. I suggest getting the coconut, but parcha (passion fruit) is also highly refreshing on a hot day. Most places have drinks as well, and there are some great bars serving specialty shots, craft beers, and Papa Jac.
This isn't an exhaustive list of things to do - and Old San Juan is just a tiny part of the enchantment that is Puerto Rico - but these activities will keep you busy until it's time for the rest of your adventure on the island. Enjoy!