By car
It can be easily accessed from Budapest by driving on the M3 highway and then the M35 about at 186 km sign highway but vignette is required. It is about a 2-hour drive. Or follow the main road 4 from Budapest or anywhere except east more than 3 hours travel from Budapest, but it is not recommended because main road 4 is overcrowded and accidents are common.
By plane
There is an airport (http://www.airportdebrece...), but commercial traffic is low. There is a discount airline flying to München, Germany, so via that Debrecen can be reached from all Europe. It is planned to extend the air traffic through involving more cheap-flight companies. Charter flights come regularly to the airport.
There are also, now, discount airlines like Wizzair that fly direct from London Heathrow to Debrecen airport, so that is another option.