Well naturally if you have been to the Hungary forum, you would know Budapest would be my first choice. Especially in late September to mid October. I think that's the all around perfect time to visit. As a matter of fact you will find me there around that time.
But since you express an interest to do more than just once city, why not do something a little out of the box. Go to Google Flights and put the cities you are considering into their search engine. But, for the destination just put EUROPE. Then filter with non-stop and set a maximum price of $150. Open the map feature and you can see all the possibilities.
At this point there are two methods. Either fly open jaw into one city, see for a few days and do the cheap flight on to your primary city (or reverse order) or set up a base in the major city and make a two or three day bonsai trip to some place you otherwise would never have considered. There are just so many place an hour flights away from every place. We do this all the time and are doing two more in 2017 along with one just long stay in Budapest. For instance your choices from Budapest include: Kiev (love Kiev), Bucharest and a couple of smaller cities in Romania, Paris, Prague, Riga, Edinburg, Brussels, Tel Aviv, Istanbul, Belgrade, Rome, Bologna, etc, etc (prices can vary a lot by date and season).
To build on Kaeleku's post below; Budapest to Istanbul is about 2 hours non-stop and as cheap as $59. Don't have a glass ball so cant speak to safety. I tend to blindly follow the recommendations of the State Department. Besides, been there, not terribly impressed, don't plan to return in the foreseeable future.
But Kaeleku does have a good point on seeing things while you can. That's why we've been focusing on Eastern Europe. The parts the tourists haven't found in great numbers still represent something very unique. In some period time all that will be gone as Eastern Europe becomes more western. Then there are the current attempts to change the borders in Eastern Europe. Parts of the Ukraine, Moldova and Georgia have already been swallowed up and the Baltic states are next on the radar.