Wow, thank you for your detailed response!
We’re considering renting a car because we'd like to explore both well-known tourist spots and smaller, off-the-beaten-path locations.
We've driven in Europe before (Italy, France, etc.), so we’re comfortable with it. We found a rental option for R5600 (~$300), which includes full insurance. Fuel prices are similar to those in South Africa, so we’re used to the cost.
We'll be arriving in Budapest from Dubai on 24 July. Our Budapest accommodation is booked from 31 July to 4 August, as we’ll be attending the Formula 1 on 2 & 3 August. We fly back late on 4 August.
Places we'd like to visit (apologies for any spelling errors):
Egerszalók
Sirok Castle
Cave dwellings – Barlanglakások / Szomolyai kaptárkövek
Götz Winery
Lillafüred Forest Railway
Hollókő
The challenge with public transport is figuring out transfers, especially since our parents (aged 68) will be joining us. They’re fit for their age, but we’d like to keep things as smooth as possible.
Tentative Itinerary:
Day 1: Pick up the rental car and drive to Eger (not as far as Lillafüred). Stay two nights and explore the area.
Next stop: Drive to Hotel Palota for an overnight stay. Explore the railway and wineries, then stay overnight at a winery somewhere.
On the way back to Budapest: Stop at interesting places along the route.
Return the car and use public transport for Szentendre and Esztergom.
Skipping Győr.
Would love to hear your thoughts on this plan!