Since you are a first time international traveler, a couple of points.
Customs in Europe is a matter of just walking through the Green Channel Nothing to Declare line. Unless you're "randomly selected for additional screening" (possible but rare), you're done; you don't talk to anybody or even have to slow your pace.
However, immigration (passport control) can take 15 minutes or 2 hours. How long it takes cannot be predicted, and unless you can convince people ahead of you to let you cut the line due to your tight connection (I did it successfully once in Nice), you just have to wait until it's your turn.
On separate tickets, you will have to go through security when going to your next flight in Rome. Note that you will use your passport as your ID for this, so keep it handy.
What I did when I booked separate tickets with a change in Rome was book two tickets. One departed 3 hours after my scheduled arrival, and another departed 10 hours after my scheduled arrival. I knew I'd use one and discard one, and the combined price was still cheaper than a last minute ticket would have been (it was a Monday morning heading to Palermo, and I could see the price of the "unrestricted" tickets rising rapidly). In the end, I made the 3 hour "connection" with time to spare - but I could just as easily have missed it.
If you can change your Rome to Brindisi flight to a later one, do so. If you check the prices for "fully exchangeable" fares and see that they are affordable to you, then buying a last minute ticket at that price can be your backup plan. If they are getting too expensive, consider doing what I did.