Things to do
Places to see. https://www.wanderingitaly.com/maps/lazio.html
Rome is in Latium too, so plenty to do there if you've never been.
Try Viterbo.
John Henderson is an expat living in Rome and he has a lovely blog about places in Lazio. Wish I had time! https://www.travelazio.it/
Am in Lazio now & have been to Viterbo, Frascati & Sperlonga. All great options.
mountain climbing
near the coast in the southern reaches
Just returned last night (Friday) after two days in Anzio. Really enjoyed the break from Rome. Fresh air, great water views, very walkable up to Commonwealth Cemetery. Had the best meal of our trip at Ristorante Turcotto - overlooking the gorgeous sea!
Quick trip from Roma Termini - hour and eight minutes, less than four bucks each way, town is a quick walk from from station.
One more note - we did an overnight in Frascati and thought it was a great time away from the Rome hustle and bustle. About 30 minutes from Termini, less than three bucks for train tix, and the Old Frascati Wine Tour was fun with the group we met.