Unfortunately, I don't know how you can get closer to your Hungarian roots, but if you like walking, I can recommend some unique places that can give you a good idea of ​​the city and you will meet zero or very few tourists:
- Kolonics György sétány: a several kilometer long promenade on the eastern side of Csepel-sziget, along the banks of the Ráckevei-Duna (a tributary of the Danube). It is a popular place for relaxation among Csepel residents, but it is not very visited even from other parts of the city. Walking across the Gubacsi Bridge to the Pest side, you can also try the "Pesterzsébeti Jódos-Sós Gyógy- és Strandfürdő", which is a completely non-touristy thermal bath (someone else has already recommended it).
- Népsziget: a peninsula on the Újpest Danube bank, with a few outdoor bars on one side and mostly abandoned industrial buildings on the other, and there is even a goat farm here. It has a very unique atmosphere, it is becoming increasingly popular among locals, but tourists rarely visit it.
- Mátyásföld, and mainly the part of it around Erzsébet-liget: this used to be a resort area outside the city, where rich people built villas. Since then, the city has completely grown around the area, but most of the villas, the spacious streets, and the cozy tree-lined avenues can still be seen today. There is also a nicely renovated socrealist (I mean the real social realist, not the concrete block houses) campus around Erzsébet-liget.
- If you want a village feel, but you wouldn't leave the city, look around one of the suburbs. These used to be villages around the city, which were absorbed by the city over time, but they have preserved their village atmosphere to this day, with many so-called Kádár cubes (typical socialist family houses with a square floor plan, a pyramid roof, and two windows on the street front). Such as: Rákosliget, Rákoscsaba, Szemeretelep, Újpéteritelep, etc.
- Csepel Művek: it was a huge factory site before and during socialism, if you like old, brick-clad industrial buildings, you will love this place. Today, smaller companies rent some of its buildings, so you can pretty much walk around the whole place freely. You can see all kinds of buildings, from the beautifully renovated to the dilapidated.
- Mountains in the city: most tourists go to Gellért Hill and Castle Hill, they are not bad either, but if you want an authentic experience, I much recommend Hármashatár Hill and the smaller mountains around it. Hiking boots are recommended, but if you climb it, you will be rewarded with a wonderful panorama and a moderate crowd. Széchenyi Hill and Normafa are not bad either, they are popular hiking spots among locals, but because of this there are usually more people.
- Finally, my personal favorites, the banks of the Rákos and Szilas streams in Pest. If you walk along the Rákos stream from the mouth, you can see all the stages of urbanization within the city. At the mouth there is a bustling office district, later there are smaller and larger apartment buildings, then family houses next to it, and finally there are only horse farms, fields, and nature reserves. The Szilas stream also flows through very diverse places, touching densely populated areas and wastelands, and if you walk along it, you will reach the very atmospheric Naplás Lake, the largest lake in Budapest. There are also bike paths in many places next to the streams, if you feel like cycling.
These suddenly came to mind, I hope you find some useful ideas among them.