Couple of questions - Austria trip

I would definitely get tickets for Schonbrunn Palace in advance. We made the mistake of not doing this and had to wait in a very long line to buy them once we got there.
For the Kunsthistoriches Museum you really don't need advance tickets.
In Europe, most train travel booked in advance will result in better prices.

The reason to buy train tickets in advanced is to take advantage of reduced rates. However, this will lock you into taking a specific train. If you simply show up to the station and buy your tickets on the spot for the next available train, you'll have no problems getting on a seat, but you'll pay a higher price. Advanced purchases can give you significant discounts, up to about 66% in my experience.

Take the Westbahn to Salzburg - no advance purchase savings

The Salzburg to Luzern could have considerable savings for early purchase

Check OBB for rates

1. Schonbrunn, probably a good idea as it is a Sunday; KHM - never had to wait more than 2 minutes at the ticket window.
2. Vienna to Salzburg - use Westbahn where you buy the ticket on the train for 24.50 Euro (free wifi); Salzburg to Lucerne - buy in advance from www.oebb.at.

Thanks everyone for the great input!