Hi Larry,
A long-term trip sounds great. As someone mentioned, unless you're an EU citizen, you can't stay in Italy a few months - 89 days. Keep the last one (or two) as a buffer. If caught, as another mentioned, you'll be fined.
It would be fun to do some things on the fly, but you should have some kind of plan in place too. I'm presuming you've not been to Rome before? Or has it been a very long time since? Things change.
Rome is a walking city; not necessary or recommended for a visitor to drive in the city. If you love architecture, you'll see some interesting & surprising buildings around every corner.
For the main attractions, everything you may want to see is within 15-20 minutes of each other - with the exception of The Vatican, St. Peter's Basilica & Castel Sant'Angelo. The Vatican neighborhood is fun to walk around. Lots of eateries & shops.
Looking at a map of Rome's city centre will help a lot. Plan in "real time" - maps make things appear closer than they are.
Yes, you can ride your bike (but it may be risky since you'll be in unfamiliar territory.) Be careful & diligent about securing it to a post. Mostly, you'll see lots of motor scooters. Lots of guys & women ride them to & from work. If you're good with crazy traffic, you can ride it all around Rome & up to The Vatican are & beyond. A bike will afford you more freedom than a car, but again...this time, the saying "When In Rome" may not be good. Driving a car, there are lots of restricted zones in the city centre.
Some suggestions:
Start to read some Italy/Rome threads. Use the search box above, put in some key words & a ton of threads will come up. You can filter down to the most recent - 6 months or less.
Get a guide book or rent DVD's. Go on YT.
Here are some how to's and intro's to Rome from The Roman Guy
https://www.youtube.com/user/TheRomanGuyTours/videos?disable_polymer=1
https://www.youtube.com/watch?v=R2gF4PoV0Ao
I like to suggest the videos because they are quite good for a newcomer to Rome/Italy in general or maybe a refresher since some things may have changed.
Sean is the narrator in most of them. Living in & leading tours in Rome for many many years, Sean has a lot of experience. There is Bojan too, another fun guy to watch. These guys are down-to-earth but quite humorous. They give simple & easy to understand tips to navigate Rome on your own.
Of course, there is Rick Steves & his video series on Rome/Italy in general.
I like to suggest watching (Italy time) https://www.skylinewebcams.com/en/webcam/italia/lazio.html You will get a good idea what all the main Roman Piazzas are like. They have skycams for some other regions too.
If you don't feel like biking it at times, there are the city buses of Rome. There is the metro, & of course, the official white taxis.
https://sites.google.com/site/sctappromeforcruisetravelers/contenuto-della-app/about-rome/using-taxis-in-rome
No tipping to any service related industry: restaurants, taxis or just about anything.
Car insurance is a good idea when driving a rental in Italy. Those who've driven in and around other regions may see this thread and chime in. Or do a search in the gray box above.
Get Travel Insurance. In case you're riding your bike & take a tumble.
Read up on carrying money (Euros); credit card(s) & Passport. If you feel comfy wearing one under your clothes, get a money belt for deep storage of extra €€€ & another card.
Don't put anything of great value in a backpack lest sneaky fingers tip toe in and take your stash. Seeabove for money belt or neck wallet.
Don't put your wallet or phone or tickets in your back or side pockets. They could fall out or be stolen; especially in crowds, on buses or metro.
Again, before you venture out, it's best to do some viewing & reading.