Jordin, your question is much too broad for any of us to answer as all of our interests are different. Many 'unforgettable' experiences are not planned but happy accidents of simply being in the right place at the right time. That was certainly the case for us, and I could not tell you how to repeat them or even if they'd mean as much to you if I could. Heck, they will not happen to US again!
Get a good guide book. Browse the forum. See what appeals to YOU and not necessarily to someone else. Then go and keep your eyes and ears open. History, art and religion in Rome, Florence and Venice? You could spend a lifetime and not absorb it all. I can tell you that with a few exceptions, our fondest memories don't have anything to do with Top 10 attractions or terrific food.
We may not remember everything in the Accademia in Florence - and we LOVE art - but will never forget an impromptu dance under the moon to a jazz band in Piazza della Repubblica one balmy night.
The meal at ___ may have been good, not fabulous, but the fascinating chat with the astronomers at the next table more than made up for it. And the couple from France than left us with their address and invitation to come and stay. And the Irish wedding party in Sorrento who made us one of their own for a few hours.
It was the unexpected room upgrade that gave us a gorgeous, wide-open view of the Gulf of Naples. O my.
It was a wander through a totally deserted Forum in Rome one very early, drizzly morning back when it was still free.
It was finding our way to the CT from Florence in the middle of a train strike. It took all day but we did it, and is a funny (now) story we've laughed about many times.
So find your moments!
Beside the obvious churches, etc
There are 900+ churches in Rome. Stop into whatever is open and you're likely to find something interesting!