Take a brolly 😬. It's not too cold, but when it rains, it really rains. Night tour of the colosseum if you can get one is fabulous, look at air B and B experiences they are usually quite good for things to do.
If you want good food at a reasonable price I can recommend this restaurant, it's not too far from your hotel.
Ristorante Mario's
Piazza del Grillo, 5, 00184 Roma RM, Italy
+39 06 679 3725
goo.gl/maps/7yNc2a5FW4b1DEqHA
Don't miss the Pantheon, it's amazing, a small group tour around the forum is definitely worth paying for, it really puts everything into context and then makes discovering everything else on your own much easier.
Bocca Della Verrita is good to see if you are a fan of Audrey Hepburn and Roman Holiday 😁
I would not recommend the bus tours, everything whizzes past so quickly, you just get a glimpse.
Have good comfortable shoes and walk and wander would be my biggest recommendation, it's an easy city to navigate and there is something amazing around every corner.
Lots of amazing gelato shops too, they are a must, this is a good one, again not too far from your hotel
The Gelatist Via Nazionale 160
Via Nazionale, 160, 00184 Roma RM, Italy
+39 06 8953 3163
goo.gl/maps/HfZ4eqqMMqP6WXpA8
Enjoy, it's the most amazing city.