Originally Posted by
Track
On the Austrian Railways website you can buy 2 tickets in first class from Budapest to Vienna and from Vienna to Prague for $168.
Not quite - you can't buy one way tickets Budapest - Vienna online on the Austrian Railways website; nor can you buy (any) international tickets online on the Hungarian Railways website.
There are no discounted 1st class tickets available from Budapest to Vienna (only in the opposite direction!), which is a bit of a pain, but not a great problem because the trains are new Railjets, and 2nd class is perfectly OK. I would wait until you are in Budapest to buy this ticket. If you can buy the ticket at least 3 days before date of travel, there are €13 special fares (including reservation). There are trains every 2 hours during the day, and although these tickets are subject to a quota, I have never had a problem getting one. Otherwise, pay the full fare, which shouldn't be more than €25 in second class.
For the Vienna - Prague journey, as
Track says, you can buy online in advance - cheapest fares in 1st class cost €39.
Regarding the OP's general question, you don't need a seat reservation on the Railjets between Budapest and Vienna, and the only day train from Vienna to Prague on which a reservation is compulsory, is the SuperCity train (EC 72) leaving Wien Meidling at 14:33 daily.