not particular for Italy, but all of EASA land, where Italy is a part of:
there are several ways, with escalating "complexity". but since you're there for longer, these may be options, too...
Option 1. find an N reg plane to charter. This is probably the easiest (if you can find one). This way you only need to be current in your FAA license, including medical and BFR. And likely you need a CFI with FAA credentials for the checkout, too. But most N-reg operators will have a few of those in the back of their hands. There was a coordinated/political push against N- reg permanently stationed in Europe a few years ago, sad and stupid story... Since this was an existential threat to the long term operation of N reg in EASA land, quite a few fled to other registers (for which you will need at least option 3 for sustained operation). Currently, judging by the number of N reg still around, some of those political restrictions must have evaporated but i know that hassles for operators have increased.
Biggest advantage in the N- reg area is easier IFR licensing/flying, so aircraft tend to be Cirri and similar high end singles and twins and with upscale avionics, hence not cheaper other than the license.
Also, do yourself and other N- regs a favor and plan several hours / flights for an FI or other local pilot to get you accustomed to local regulations, airspace, administration of landing fees, border crossings etc... Not really hard, just different and you don't want to raise eyebrows and stand out as "the typical N-reg"...
Option 2: "Holiday license"
just read about this for the first time, there seems to be a new 2020 european regulation allowing to use a foreign license for a maximum of 28 days with a simple checkout with an FI
may be an option for other visitors but likely too restricted for your lengthier stay.
Option 3: "Validation" based on FAA license.
Here, you need to keep the FAA license valid in the background (minus the BFR, i believe, but that is replaced by the bi-annual EASA flight with a CFI).
this is probably the way to go if you stay longer. You're then allowed to operate any EASA registered airplane in any EASA country, including non EU states like Norway and Switzerland.
I believe it is even possible that some of that could be done from the U.S. / Some U.S. flight schools specialize in European licensing.
Just be careful that most U.S. flight schools have dealt for this via the UK CAA, which due to Brexit has become a very unstable path, also for those, who already have validations via the UK.
Minimum is 100hrs flight time (only an issue to newcomers) and a skill test (EASA speak for check ride with an examiner, not just an FI) and obviously some time required for paperwork.
Another thing here to consider is that although the licenses and medicals are valid across all EASA land, you kind of "get locked in" with one authority and need to have e. g. the medical and the license with the same authority, and transferring from one to the other can be a hassle. So you should choose your "base" wisely, and possibly even not necessarily Italy. This is a total idiosyncracy and against the idea of one medical and one license. Especially for people developing medical trouble this can make a **** of a difference.
Option 4: "Conversion" of the FAA license to EASA.
This is the most extensive/complex undertaking and in the end is essentially a completely separate license/medical. For private flying overkill in my opinion unless you stay in Europe for much longer or aim for commercial/airline gigs etc...
It requires theoretical exam in air law and human performance, 100hrs minimum, an EASA medical and a full skill test among other things.
Sounds practically the same as Validation but the hassles are significantly more.
Driven mainly by two topics:
Access to theoretical exams is typically administered through flight schools, not just an easy online test you can just signup (that doesn't say that this couldn't ve offered somewhere) . So they will want to sell you theoretical lessons and make sure their candidates pass the tests.
And the medical, for reasons mentioned above. Locking you in to one country's authority besides being more expensive and thorough than the FAA medical.
Option 5: Ultralights
Italy has a huge ultralight scene and getting an ultralight license should be relatively easier. But this is nationally managed and language could be an issue there. But if you're just urging to get a bit of air under wings while in Italy, that may be sufficient. The classic GA has a tough position in Italy due to fuel prices and the infrastructure which has either big and expensive airports or then small "aviosuperficie" more suited to the lower end of the scale...
Option 6: Wait and speculate
There was a bilateral agreement between the FAA and EASA about 2 years ago that should pave the way for much easier reciprocal acceptance of licenses etc... First area where i've seen that applied successfully was with avionics (TSO/STC stuff). There was a plan to also include PPL etc... But the whole 737MAX clusterf....k seriously threatens all advances in that area :-( But that may possibly bring a sudden path/flexibility which is not yet existing...