Everyone has their preferences so I imagine you will get all sorts of suggestions. So I might as well through in my preference.
October is the perfect time of year to be in Central Europe; and the best of Central Europe is Budapest. It's not hot, the evenings can be a tiny bit cool, the outside cafes are all open day and night. You will almost always find me there the first week in October.
A quick tour: https://www.youtube.com/watch?v=dZYpHr1wDmY
Bourdain: https://www.youtube.com/watch?v=0nd9DuDGCz0
Just cause I like it: https://www.youtube.com/watch?v=cI6fb6olbZI
Fortunately Budapest isnt buried in tourist yet. Well outside of District V at least. A little like Paris where the tourist sites are spread fairly broadly over a fascinating city fabric; the places in-between can be the most amazing. The "real" Budapest. Sounds like what you are looking for. Better than Paris, public transportation is for the most part above ground on classic (and a few new) yellow trams. While the person in the seat is going to work, you are enjoying your "tour bus" with huge windows so you miss nothing. And with a cheap Travel Card you can hop on and hop off at your hearts desire.
Because you want to meet the culture, stay in District VI within a block or two of Andrassy ut. Andrassy ut is the Champ De Elysees of Budapest and the historic millennium metro line runs down the center of it (M1) taking you from the river at one end to the City Park and Bath House at the other. Along the way are Tram connections to reach every corner of the city.
To get more local, rent a short term apartment. Shop at the local grocery, drink at the corner wine bar and say good morning to your neighbors, Take a cooking class, go to a service at the local synagogue (being Jewish not required), go to a theater full of locals (as opposed to tourists), ride the bus and get lost, go to a jazz club, …...
https://1drv.ms/v/s!Ai7Zk-szxfTJhetCiVLDEf1GfYSX4Q
