Welcome to Flyertalk, alhw03.
Well, the price for all of you by train would start at €87 for a pre-booked, inflexible ticket (€29/adult). Like airlines, availability at the cheapest tariffs is restricted. You can book online at
http://www.bahn.co.uk/ but occasionally see better fares at
https://ticket.oebb.at/bin/frame_tic...ket=jticket.pl - those are the sites for German and Austrian railways respectively. Both have e-tickets available.
You should be able to buy through fares, that include the S-Bahn (commuter train) connection from the airport to Munich Hbf - this is a simple journey, covered by a couple of S-Bahn lines, but you do need some time to change from the underground (commuter) section of the station to the main lines on the surface.
If you can, it's worth getting one of the many, direct Railjet services from Munich to Vienna. These are excellent quality trains, and save you changing (you can tell they're Railjet as the train number will start 'RJ'). If the timing doesn't work, however, a 6-minute connection may be alright, it depends on the station you're changing in. It's certainly doable at smaller stations (where it may be the same or an adjacent platform). However, if you've booked an inflexible ticket (on one itinerary) with a connection that you miss due to delayes, you will be allowed to travel on the next available service.
It takes about 5h15 from Munich Airport to Vienna this way.
The Railjet services arrive at Wien Westbahnhof, on the western edges of the city centre, and so a good point from which to take an onwards U-Bahn (metro). If you connect to a different service, you may arrive at Wien Meidling, a temporary terminus while the Sudbahnhof is rebuilt. In that case, you can generally book a free S-Bahn connection onwards to Wien Mitte (or wherever is most suitable) or use the U-Bahn.
By plane, you have the advantage of already being at MUC (although that means you need to get from VIE to Vienna - easy enough via S-Bahn or CAT train). FlyNiki operate MUC-VIE, generally around the €50-80 mark per person (child discounts only up to 11) with a few weeks' pre-booking, although you can get down further if you're lucky. The flight is an hour, so it's going to be quicker than flying, door-to-door.
In both cases, of course, you'll need some contingency time to allow for flight delays, as you won't be covered for the onward journey to Vienna if you're late. It's possibly worth planning something to do in Munich in case you're left hanging around for a couple of hours!