We will be traveling from Prague to Salzburg, Vienna and Budapest in September by train. Is it better to purchase train tickets ahead of time (while still in US) or wait till day of travel?
The further out you buy your tickets the cheaper it is. I don’t see direct trains from Prague to Salzburg but this website will allow you to buy a ticket: https://www.cd.cz/en/default.htm. The Man in Seat 61: https://www.seat61.com/ will recommend what websites to book your other tickets on.
If you buy your train tix now on the Czech Train website provided in the above post, the Prague to Vienna (5hrs) ticket can be as low as $16. Prague to Budapest can be as low as $15. You must use the Czech website to get those prices for trains originating in the Czech Republic. For Vienna to Salzburg and Vienna to Budapest you can see train schedules and current prices at www.TheTrainline.com
An even better place to look up schedules and prices for Vienna to Salzburg and Budapest is www.oebb.at