Do I need to purchase train tickets Prague to Vienna to Budapest to Zagreb?

Advance tickets tend to be less expensive but lock you in and further reduce travel flexibility. Rarely do trains sell out so that is generally not a reason to pufchase in advance. I think the best is to buy tkckets for leg two at the train statiln when you arrive on leg one. This will give you a fes days advance purchase and an open ticket. Still, ig will cost a little more ......

You may find an advance discount for the two routes that touch Austria, if you lock in departure times, but full fares are relatively affordable to buy in stations. There are only a couple of direct Budapest-Zagreb trains per day but I see no benefit to ordering on the Hungarian web site (for pick up at the station).