Just off the cruise ship there's a small strip of town dead ahead (a long east-west point), some good souvenir shopping mixed in there. We've walked from one end of that point to the other. Found some nice cafes hidden in some of the streets we walked, like one where we got some great Greek baklava ice cream. On one end of that point are a couple of nice "castillos" (forts) that are nice to see, right on the shoreline, plus some pleasant scenery along the Paseo de los presidentes, the Capitol area, etc. It's pretty along there.
But just to the left off the ship, usually there are (or used to be) some smallish tour buses and taxis so you can go south into the main part of the city. Last time we were there we got a 2-hour bus tour (with a couple of stops) for $10-15, can't remember exactly how much. But I think there used to be "city" buses that you could hop on and hop off.
Expect it to be hot, especially on narrow streets where some of the best shopping might be. As others have implied, San Juan is really one of the best walking ports we've been to. And yes, safe and clean.
Edit to add: I might've skewed the directions a little bit. But if you have any kind of a map it isn't too hard to figure out San Juan proper.