March in Budapest, you have a couple of months before you will regret not booking a hotel. Opera tickets (if that interests you) you need to do now. The other theaters vary a bit but you need to start checking now who has 2027 shows listed.
If you only have six days, I might look at bus options from Budapest, maybe Budapest-Bratislava-(Vienna), Budapest-Zagreb, Budapest-Belgrade.
Everyone has their favorite way to travel. Ivan is correct, sometimes buses can be interesting as they pass through small towns while the trains whiz past them.
Belgrade isn’t one of my favorite destinations so I will let someone else explain the beauty of visiting. Bus or train its about 6.5 hours. Once thing to be aware of is you will have to go through passport control and that could take time.
There is a fast bus to Zagreb in about 4 hours. The best train is 6 hours, but I can get up and walk around on a train, so I would still vote for the train. But I think Zagreb is best saved for that June trip (early June is ideal in my mind). Just to keep it all in one place; on June 7 in Zagreb, the average high temperature is 25°C (77°F) and the average low is 14°C (58°F). This creates a pleasant daily average of roughly 19°C (66°F). In June the city averages about 8 to 9 rainy days for the month and in March about 8 to 13 rainy days.
Budapest – Bratislava – Vienna, would be hands down on trains in my book. But others might like the bus for all the stops and the towns along the way. Never know. It’s a fair reason to ride a bus. I imagine about the same time as the train. But the OP has already done Austria and even if the OP missed Bratislava, it’s not a place I would go out of my way for.
You are doing Croatia on another trip in June, but after you fall in love with Budapest you might return in June on the way to Croatia, so, for that trip (really this is for anyone else looking to do a similar combination) I suggest you fly from Budapest to Podgorica, Montenegro. The flight takes 1:15 and can be as cheap as $36 but count on closer to $75. So many who go to Croatia ask about day trips to Kotor. I sort of think it’s a lot of driving back and forth, so this allows you to start at Kotor then work your way up the coast to Dubrovnik. No back tracking. I would stay in Montenegro (maybe Dubrovnik too) but I am terribly biased on the subject and not a good source.