I forgot to answer the first question!
Innsbruck- Salzburg has a fast train every 2 hours, usually Railjet design (which most Austrian long distance trains are now), there were a few odd services where German ICE trains appeared under a joint usage deal. Best prices are online, weeks in advance at
http://www.oebb.at/en/
The fast trains take a short cut across a corner of germany without stopping and takes just under 2 hours, the germans currently have some work ongoing so some trains may be altered . The roue that stays within Austria is via Zell am See and takes longer, this has the scenery though, and all days has regional trains.
If you book the Railjet in advance you might get it for €28 each, this is a fixed booking and prices go up as seats sell. But, if you travelled via Zell using regional trains you could buy an "Einfach Raus" ticket for €33 for two people http://www.oebb.at/en/angebote-ermaessigungen/einfach-raus-ticket this is not valid beofre 09:00 Monday- Fridays, which means the first worthwhile train from Innsbruck Hbf (main station) is at 10:35, change trains at Wörgl, arrive Salzburg at 14:51. You double the jounrey time but could halve the cost - and you get the scenery. This ticket can be bought on the day of travel at the station. It is actually a day pass, so you can use other local and regional trains for connections if needed.
Salzburg city is actually quiet compact so you may not need transport
Vienna is definately not compact, the local trasnport authority sells passes for 1, 2or 3 days https://www.wienerlinien.at/eportal3/ep/channelView.do/pageTypeId/66533/channelId/-47345 the 1 day is €7.60