Budapest-Vienna-Prague-Budapest Triangle Pass?

We made this route 20 months ago, and the trains were not high-speed trains. They were old and slow between these cities. We were in second class and the a/c barely worked in April. I would have gladly gone 1st class had I known the a/c worked better. And who knows if there will be refugees on this rather popular route to the northwest and Germany.
If you're going in and out of Budapest and you're limited to just over a week, forget Prague. Budapest and Vienna should provide you with more than enough tourist sights to take all of your time. Budapest is a ball with all the grunge bars and nightlife.

You need at least 10 days to do those 3 cities. If you have only one week do just two. You don't need reservation ahead. Just buy ticket when you get there. 1st class cost 50% more and really it's not needed. There is no real high speed train on that route. I never traveled between Vienna and Budapest, just between Prague and Budapest (which goes via Bratislava) and Prague and Vienna. Speeds of trains there are between 100 and 160 km/h = 60 to 100 mph so not as fast as France TGV or Germany ICE. If you reserve specific train and seat and don't show up you lost money for that specific train. That's why I would not bother to reserve. It is mandatory for some trains to reserve seats. On some other trains you just buy ticket but if it's full you stand. However you can buy a seat ahead if you want and whoever is sitting on your reserved seat would have to leave it. If I suspect train could be full I reserve seat. I am not familiar with the triangle pass but I think point to point would be cheaper.

I am guessing you are doing 9 days (Saturday to Sunday)
Saturday Leave the US
Sunday Arrive Budapest
Monday See Budapest
Tuesday See Budapest
Wednesday to Vienna
Thursday in Vienna
Friday in Vienna
Saturday to Prague
Sunday See Prague .................................. ohhhh no Mr. Bill ............... you are out of time!!!!
Monday see Prague
Tuesday Prague to Budapest
Wednesday return to US

Stick to something similar to this
Saturday Leave US
Sunday Arrive Budapest
Monday see Budapest
Tuesday see Budapest
Wednesday to Vienna
Thursday see Vienna
Friday see Vienna and late train to Budapest
Saturday see Budapest
Sunday depart for the US

I would leave Prague out of this trip with the amount of time you have. Budapest and Vienna are just a 3 hour train trip apart and I made this trip in 2014 very easily, 6 days in Vienna and 4 in Budapest did justice to both. I loved them both much more than I did Prague, visited there this past May for 4 days.

Very easy to buy train tickets, for a 3 hour trip I did second class and it was fine. I used the OBB website since I originated in Austria, you just take your confirmation to a kiosk and print your ticket at the station.

Regarding the use of the Triangle Pass: It is only offered for 2nd class travel ($144), not 1st class. Most of these trains don't require seat reservations, but you can make optional reservations (in 2nd class) for about 4 euros each when you are there or $11 when you buy the pass. Although schedule sites don't say so, I've recently been told that the RailJet does requires reservations in Hungary (your first leg), but there would still be no deadline to book those. The reservations are not an integral part of the pass; if you miss a train, you just loose the reservation fee. The opportunity to hop on any train on this designated route is the flexibility benefit of the pass.

On the other hand, if you book advance-discount tickets, they will be restricted to particular dates and times (even if the seat reservation fee is still extra) and you'd have to work through more than one national web site (i.e., Austria and Czech Rep) to book all three. Full-fare tickets purchased on short notice would add up to about $195 in 2nd class.

Budapest to Vienna can be as little as 13 euro if purchased in advance. Don't get too hung up on "RailJet" between Budapest and Vienna as the standard EC trains aren't any slower. Select by schedule. We've done both 1st and 2nd and it really doesn't matter that much..

You may want to consider a direct flight between Budapest and Prague on Czech Airways. Last summer it was only about $100/person (w/exchange rates that existed then), and took less than 1.5 hrs, as opposed to an all day or all night train.

In reality plane is only slightly faster than train:
You need about 1 hour to get to the airport in Budapest (just in case because of traffic). You need to be at the airport 2 hours before your flight. Flight takes 1.5 hour. To deplane, collect your luggage and taxi from the airport to your hotel 1-1.5 hour.
By train: taxi to the station at most 0.5 hour, train 7 hours, taxi to your hotel in Prague at most 0.5 hour.
Plane: 1+2+1.5+1.5=6 hours
Train: 0.5+7+o.5+ 8 hours. 2 hours difference. Is it worth the money, inconvenience? Besides you can sightsee on train.

I think you are about a half hour shy on the train and about a half hour long on the plane. So closer to a 3 hour savings for about $100. Pretty worth it for me. But i can understand either opinion.