I've done this trip in April this year so I can share some of my observations here. I took an early evening train from Szeged to Subotica, stayed there overnight and continued on the 15:05 train from Subotica to Novi Sad (via Sombor). There I reserved a seat on one of the high speed Soko trains for the next day to get to Belgrade, which do the trip in just 36 minutes.
First of all there were major construction works going on in Subotica so some sort of emergency platform was in use which handles all the train traffic, situated a bit to the south of the station with the entrance being practically opposite the Lidl supermarket on the road called Pala Papa, it's not well marked. On the platform there is a small container functioning as the ticket office.
The Soko trains indeed have compulsory reservations, and they do sell out but they are very frequent. Usually there are still enough seats available up until about an hour before departure, after that I noticed the seats sell out pretty quickly. I remember being able to see how many seats were still available online, but I can't find that info anymore so they might have removed it. There are also relatively frequent regional trains from Novi Sad to Belgrade using FLIRT trains which take only about 20 minutes longer, don't have seat reservations and can't sell out.
Belgrade Centar station is thankfully now more than a "minor halt" with a pretty nice new station hall having now opened with much better amenities than what it used to have, such as a few kiosks (where I was able to buy a day ticket for the public transport in Belgrade), a tourist information counter, new ticket offices and an excellent and affordable restaurant, where I enjoyed a nice meal after arriving from Novi Sad and before taking the night train to Bar. The only drawback is that the station is still poorly connected to Belgrade itself by public transport with only a bus connection, tram connections are available at the Novi Beograd station where all trains coming from Novi Sad stop as well, which has also received a major overhaul and doesn't look as run down anymore as it used to. There also tram connections to the Karađorđev Park and Vukov Spomenik stations on the BG:VOZ (run-down kind of S-Bahn) network.
It's possible to do Szeged-Belgrade (or even Budapest-Belgrade) in a single day, taking the 08:58 train from Szeged to Subotica, arriving at 10:24, then leaving Subotica at 15:05 (yes, there is no better connection) arriving at Novi Sad at 18:41, then take one of the high speed Soko or regional trains to Belgrade with the last Soko train leaving at 21:30 and the last regional train at 23:50.
There is also a very early train from Subotica to Novi Sad at 05:22 in the morning, and from September 1 there is also an additional possibility of getting to Belgrade via Zrenjanin, taking the 07:06 train from Subotica to Zrenjanin arriving at 10:11, then taking the one and only train from Zrenjanin to Belgrade at 10:55 arriving into Belgrade Centar at 13:09. This train does not stop at Novi Beograd but it does stop at Vukov Spomenik and Karađorđev Park.
