How do I get from Budapest to Sarvar using hungarian public transport ( Cheapest )
Normally the trains to Szombathely will stop at Sárvár station just 20 or so minutes before Szombathely, depending on the train. The cheapest domestic train fare is 3010 HUF. But depending where you coming from and do you want to go round-trip or not you can "shave" a little bit off.
For instance, Budapest-Sárvár-Budapest is 6020 HUF. But Sturovo-Budapest-Sárvár-Budapest-Sturovo international round-trip train ticket bought at MÁV international counter is 22.24 Euro, you will pay 5782.4 HUF instead of 6020. So there is no exact answer (yet) because I need to know a bit more, than what you told.
Off-topic: consider to visit not just Szombathely but also the monastery of Ják, close to Szombathely.
Just take the train. Direct trains bound for Szombathely call at Sarvar also, except for InterCity services. Fare quoted is correct, 3010 HUF one way. I say don't bother with international tickets this time for those few hundred forints(that will take you much more time than buying a domestic one). Otherwise, GyuriFT knows good tricks how to save on train tickets!
Vadkert krt. 5, Sarvar
Varkerulet 2, 9600, Sarvar
1 Rakoczi Ferenc 1 H-9600 Sarvar
Vadkerti ut 4, Sarvar