Well, you have a number of possibilities to travel relatively cheap. Let's look at each leg separately:
Frankfurt-Vienna: easiest and fastest (if we speak about 'by land' journey) way will be to visit
DB (German Railways) website (there you also find all the timetables you need, both for Germany and Austria) and purchase in advance discounted tickets for the direct train (numerous connections per day, including overnight). If you do so good in advance (ticket sale starts 92 days before departure), then the fare can be rather low. In case you travel date is soon or there is no good deal on rail tickets for the connection you need, then you can also take advantage of German (depending on day of the week it will be either
Schönes-Wochenende-Ticket or
Quer-durchs-Land-Ticket) and Austrian (
Einfach-Raus-Ticket) group travel passes. More info about those on our
Germany and
Austria pages. The only inconvenience is that such passes are valid for regional trains only, so you won't be able to use direct ICE train Frankfurt-Vienna, but will have to change trains and travel time in totall will take longer. Imho, with such group passes it might make sense to do Frankfurt-Salzburg-Vienna-Munich, thus to avoid long leg Frankfurt-Vienna.
Also, in case you aren't specifically up for a train ride, there are relatively cheap flights between Frankfurt and Vienna (for instance kind of low-cost company FlyNiki)
Vienna-Salzburg: three options here: a) use mentioned above Einfach-Raus-Ticket, it will allow your whole family to travel from Vienna to Salzburg for only 32 euro in total (although again train change will be required). Einfach-Raus-Ticket can be bough even on the day you plan to travel, so no in advance purchase necessary; b) you can also buy discounted ticket for the direct intercity trains (start from 19 euro) on
Austrian Railways (OeBB) website, here in advance purchase necessary, since cheapest tickets get sold out quickly; c) don't bother with any in advance ticket purchase, but just use private company -
WestBahn, tickets can be bought even on board of the train
Salzburg-Munich: here your best and pretty much the only reasonable choice (taking into consideration number of people traveling) is Bavaria Lander Ticket (a regional pass for Bavaria, which is valid till/from Salzburg also)
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