I live in Bratislava and have been to Sofia but not Bari. First, make sure not to call Bratislava 'Eastern Europe' 8n front of the locals, as they are insistent it is Central Europe.
It's a question of how long you have. Bratislava doesn't need more than two days. If you are looking to do something different, why not really explore a single country rather than try to hit as many capitals as possible. This would be my suggestion for a Slovakia tour. You will have a great time and see places none of your friends have been.
Day 1: Fly to Vienna and take the bus to Bratislava. If you haven't done Vienna, spend a night and day there to get the highlights.
Day 2: Bratislava. This is a really charming city and can be seen in a day. Try to make it out to Devin Castle to see crazy picturesque ruins and enjoy the local wine.
Day 3: Pick up a rental car early and hit the road to Trencin. What's there? Oh, just a massive hilltop fortress dating back to Mattias Corvinus and inscriptions on a cliff from Marcus Aurelius' legions guarding the Roman Limes. Clamber around the castle and have lunch if you hit the road early enough. Then drive about 1.5 hours to Bojnice and take a tour of their Romantic real-life Cinderella castle. Overnight in Bojnice at a spa hotel.
Day 4: Drive to the medieval mining town and UNESCO site of Banska Stiavnica. Make sure to have dinner at the brewery's restaurant, drinks at the ruin bar, tour the old mining shafts, and if you have energy, climb up to the "Calvary" to take in views of the volcanic caldera the town lies in. Overnight at one of many B&Bs.
Day 5: Up to the High Tatras, book a room at one of the hotels around the Alpine lake off Strbske Pleso. Take a day hike into the High Tatras national park or just putz around Strbske Pleso and take a chairlift into the mountains.
Day 6: Into the East. Stop and tour Spis Castle, the largest castle complex in Europe. Drive into Kosice and stay at one of the boutique hotels. Tour the massive St. Elisabeth Cathedral (the farthest East Gothic Cathedral in Europe). Eat at one of the restaurants offering local 'Eastern' grub and brandy. Marvel at the fact that the women are even more beautiful than in Bratislava, which seems impossible.
Day 7: Head to the southeast Tokaj Wine region and stay at a B&B/ winery. Or, take the three hour drive into Budapest. Either way, hit Budapest on your way back and budget two nights at least. Make sure your rental company knows you are taking the car into HU. Drop the car back into Bratislava when you are done with it.
If I was expanding this, I'd budget more time for Vienna, Tatra hiking, or Budapest depending on preferences. You will be amazed at the affordability of rooms, food, and the car gas will be the big ticket option.
You'd be amazed at what there is to explore. Google a few of the sites I suggested and take a look.
