linda,
I'm not sure a car rented in Italy will be the best way to see those countries. I'd recommend that you have a look at the State Dept. Travel Advisories before booking anything - https://travel.state.gov/content/travel/en/traveladvisories/traveladvisories.html/ (click on the "Travel & Transportation" tab). The road conditions in some countries are less-than-ideal and some of the warnings for Albania were especially serious.
There could also be insurance implications when driving an Italian rental in some countries. If your drive from Italy to the Balkans takes you through Austria, you'll also have to buy the Highway Tax Vignette as soon as you enter the country. Failure to display a valid Vignette can result in very hefty fines, collected on the spot!
You may already be aware of this but one important point to note is that each driver listed on the rental form will require an International Driver's Permit, which is used in conjunction with your home D.L. These are easily obtained at any AAA / CAA office for a small fee and are valid for one year. These are compulsory in Italy and some of the other countries, and failure to produce one if request can result in hefty fines collected on the spot! For driving in Italy, you'll also need to be extremely vigilant to avoid the ZTL (limited traffic) zones which exist in many towns. Each pass through one will result in hefty fines which you may not know about until months after you return home. There are also speed cameras, tolls, etc.
Given the countries you plan to visit, I'm not sure this will be a carefree driving holiday. It might be easier to rent vehicles separately in each country and use well planned public transit for the remainder. I'd suggest contacting www.gemut.com either by E-mail or on their toll-free number. They have a lot of experience in European car rentals and will be able to provide you with some good information. Gemut offers a free downloadable PDF rental guide on their website. This is one section of the guide pertaining to Eastern Europe....
"For “Zone two”—countries such as Romania, Bosnia, Serbia, et al—plan to pay substantially more. You may also be required to rent the car for a minimum period, pay “cross-border” fees, and purchase extra insurance."
According to their rental guide, in some cases, rental firms will only allow manual transmission vehicles to be taken to Eastern Europe. If you're not able to drive a manual, that will be a problem.
While many people on the forum may have driven in those countries without any problems, this is probably one trip where it will be very important to "do your homework" prior to the trip.
One other suggestion to consider, is to use the trip planning service offered by Rick Steves Europe. There is a small fee for this service but the well travelled staff may be able to offer some good travel information to help you have a trouble-free holiday - https://www.ricksteves.com/travel-help/plan-your-trip-with-our-expert-consultants .