How to get from Belgrade to Romania/Hungary?

There is a daily international train from Subotica to Kelebia (Hungary). From there you have to take a bus/taxi to Szeged. 

They have. There are daily trains from Subotica to Novi Sad. From there are every hour a direct train to Belgrade:

Budapest Nyugati                  10:53
Szeged                       13:15  14:22
Kelebia                       15:17  15:41
Subotica                     15:55 
stay overnight in Subotica, or take a direct Bus to Belgrade
Subotica                                05:05

Novi Sad Ranzirna     09:10   09:10
Novi Sad                    09:10   10:00
Belgrade Centar        10:36

Szeged - Kelebia and Novi Sad Ranzirna - Novi Sad you can make with rail replacement service

PAYable (not very exp.-certainly not for FI) buses Sub→ Bgr run ev 1 or 2 hrs-from, what else the BUS station (=other side of town as train) or in Bgr from the long standing BAS (Bgr Autobus St.)

In HU they completely rebuild this line-since a few monthes. The route now advised is normal IC to Szeged (has also a section NONreserved=non AC), then a few times/day rail-replacement bus to/via kelebia to Subot.-check mav-start planner. Pass accepted on this bus.

There ARE also direct buses Bgr-Bp (probably volan=the main HU state-bus), or try flix.

That train above to Sub from Bgr=serb side makes a huge detour and is dead slow-and some say also often cancelled (the trainset is broke again-happens very often there-after 30 yrs of non-maintenance)

NSad-BGr is since a few weeks open again with brand new trainsets and for now takes NO passes-but is only a few€=promotional introduction price. The payable bus from Sub runs along it and avoids the new change. Note that the old main station in Bgr (maybe still in your guidebook) is NO more-its a bleak suburban stop without even a good bus/tram into town