Cabo Rojo has a bioluminescent bay, but it has lost part of its shine due to pollution. It’s still there, though, but probably not as “ooooh” as it once was.
I’m just glad you’re spending time away from Old San Juan. I mean, as much as I love that area, I would prefer people to go outside the city and look around. Personally, I enjoy just roadtrips and looking at the scenery.
Still, I’d recommend spending a morning or an afternoon in Old San Juan. Visit the forts, see the old buildings.
Since you’re in the “isla” and not the “área metro”, I’d say visit Ponce. It has historical buildings, some tourist areas around like Castillo Serrallés, Hacienda Buena Vista (coffee plantation), and Parque Ceremonial Tibes (Taíno and earlier arqueologic park).
For museums, there is the Museo de Arte de Ponce in the south (the famous painting Flaming July is there), and the Museo de Arte de Puerto Rico in Santurce (part of San Juan).
Other nature-type places you can check are El Yunque (rainforest park), and Cabezas de San Juan (in Fajardo, a mangrove park).
In the west (around where you will be in Mayagüez), there is the Faro in Rincón and parque Guajataca in Isabela/Quebradilla.
Check around because I’m not sure if some of the places I mentioned will be open, or if they require prior reservation.
Also, go to the beaches. I’m partial to the west ones (most of all Crashboat, in Aguadilla).
I’m not sure how you can get around pitorro, as I’ve never tried (I don’t even like normal rum).