By suburban railway (HEV in Hungarian)
It is relatively easy to reach Hungaroring by public transport. Take the HEV railway from Ors Vezer Square and alight at either Szilasliget or Hungaroring itself, journey time around thirty minutes. The former has a paved walkway to the track, whilst for the latter you will walk over a dusty road.
Ticket holders for the Grand Prix can board a special bus laid on for the purpose. The bus departs from the Pest end of the Arpad Bridge, and takes you to the centre of Mogyorod, the village adjacent to the track.
Train seems the best option, as I assume you can avoid traffic going to the site. But I suppose it could be busy.
I ended up taking the bus on Saturday and Sunday. This wasn’t a problem getting to the track either day, or getting back on Saturday, as people are coming and going so the travel pattern is spread out. Sunday was a nightmare, though. Everyone trying to leave at once, and the heat was intense. One fan passed out, and had to be pulled out of the crowd by his friends. People were lined up 10 deep for probably a couple hundred of feet along the road. The buses were stopping in random places, and everyone would just crush in. There were no lines.
If I go again, I think the bus is fine for Friday or Saturday, but would probably look at another way back on Sunday afternoon.
I suggest to use the train on the way back next time. The walk is a bit longer, and it is not free, (you have to buy a ticket but it is about $1) but trains go from the circuit at every 15 mins on Sunday, and are a tad bigger than the bus. On our way back, we just missed the first, so had to wait a while, then the trip is about 30 mins back to Budapest – Örs vezér tere.