I have never been to Debrecen, but I have friends who have lived there and say its quite lovely. You can find the sites of interest with a few Google searches. "undoubtedly want to spend a couple in Budapest" in my most objective point of view (not always easy) I would suggest that it takes a good solid three full days to appreciate Budapest. Put more accurately wake and go to bed in Budapest three consecutive times. I haven't rented a car in the region so I don't know how willing a rental company will be to have their car in Ukraine or even Romania for that matter. I have driven into Slovakia in a rented car with nor problems. I have been to Romania and Slovakia (you might consider Bulgaria too) and they are worth the time if you have the time. You can do a lot in Slovakia by train out of Budapest, cheaply and conveniently. Romania is a little more difficult but fortunately the services in Romania are much less expensive. You can hire a guide, with a car and pay for a decent hotel room for something between $300 and $400 a day; and he will pick you up in Budapest ( and possibly take you through Slovakia and Ukraine as well). I have been working on a fishing trip for late next spring that will be in Slovakia and Ukraine. What I have determined is that transportation from Slovakia through the Carpathians and then back to Budapest is somewhat limited and even in private cars short distances can take long periods of time. For me it adds to the flavor so I'm good with it. Also consider Romania and Bulgaria as they complement each other quite well. This year we fished across Bulgaria from Sofia to the eastern border then crossed into Turkey, spent a few days in Istanbul and flew out of Istanbul. Air rates to and from Istanbul to the States or to Budapest are very reasonable.
Check out this guide for Romania if the idea seems reasonable: Daniel Gheorghita, http://covinnus.com/
This guide for Bulgaria: Pavlina Docheva, Easy Bulgaria Travel Ltd, www.easytouristguides.com, email@example.com
In Budapest: Alíz Tóka" firstname.lastname@example.org
In Hungary: Andrew ILLES, http://www.guideinbudapest.com
Does all that help you any???