Hi there...we are doing Rome on our own too. It seems like I have spent hours researching Rome. You are so lucky as you already know where everything is. We plan to walk to the train station and will be doing St. Peter's Basilica first like you. (We have tickets to the Nacropolis at 10:15). Then, we plan to do the Vatican...not a tour, just walk through. We will be traveling with our 8 year old dd and a 12 year old ds and I know they do not want to spend 3 hours inside the Vatican. However, your 3 y/o may be more patient than our 12 y/o. The HOHO bus sounds good and I also heard that taxis are pretty reasonable in Rome. Good luck!
Rome isn't too hard to figure out if you have a map -a lot of things are walking distance from each other. We walked a lot last time vs taking public transit. But I didn't have a toddler with me last time! We took a taxi a few times, my only advice is that make sure they go by the meter - because some taxi drivers will try and quote you a grossly higher amount before you leave. Taxi stands are everywhere to catch a taxi. I always would hop in and state my destination confidently like I do it everyday and then they would just go by the meter - which isn't too expensive, but certainly more than the metro.
You are so lucky to have tickets to the Nacropolis! I really want to do that and heard its amazing - but they don't allow children.