Been there many times, and have a winter condo there. This is probably the best web site for tourism:
Puerto Rico Vacations | Puerto Rico Tourism Company | Puerto Rico Tourism & Travel
As far as what to do, you can do just about anything except snow ski, and there is probably someone trying to figure out how to do that. Here are some ideas:
Old San Juan. One of the oldest cities in the New World, dates back to 1520's if I recall. It was a walled city and there are still city walls that you can see. Go to see the old Spanish forts El Morro and San Cristobal, run by the Park Service. Walk around town, it's a bit like New Orleans with the balconies up, cobblestone streets (brought as ballast in spanish galleons), nice restaurants. You don't have to speak spanish, everyone speaks english but if you want to speak spanish you will have a leg up, except as soon as they detect an accent they will speak english back to you.
The rain forest at El Yunque, the only tropical rain forest in the US Forest Service. The peak is at around 3500 feet, gorgeous view of the island and the ocean. Worth at least half a day, and about 2 hours east of San Juan.
El Conquistador Hotel on the east point of the island is very impressive, sits high on a cliff overlooking offshore islands, beautiful golf course and tennis courts, but you have to stay there to use them.
The offshore islands of Vieques and Culebra. Vieques has the best example of a phosphorescent bay, but must be seen on a moonless night. Culebra has one of the top ten rated beaches in the world according to Travel magazine. You can fly to those islands, or take a ferry for less than $5 from Fajardo.
There are so many other things, but on a first time limited vacation they would be too much to do.
As far as slums yes, there are those and they are almost as bad as those in downtown Washington DC
But not many tourists go looking for them.
As far as retirees there, no there are not many and I think it's mainly because of the cost. It's a US style economy, and prices are a bit higher than on the mainland because everything has to be imported. But we know several US retirees there, and DW and I do a lot of sailing adventures either alone or with them. PR is a great jumping off place to sail and discover small uninhabited islands, or sail to the British Virgin Islands. As I say, we are only there in the winter because we want to avoid hurricane season. And even though I speak Spanish pretty well, DW doesn't so she gets a language headache after a couple of months
Good luck and let me know what you think after you return. PM me if you want any details.
Here's a link to the phosphorescent bay
Island Adventures Biobay Tours, Vieques Island - Puerto Rico
And here's a link to Flamenco Beach on Culebra island
Flamenco Beach Culebra, Puerto Rico