Is there a local bus that serves Garmisch from Innsbruck or is it only train?
Why would you want to ride the bus when there is a train? I know there is an RVO bus (www.rvo-bus.de) from Garmisch-Partenkirchen to Mittenwald. I've seen it on Bahn schedules. According to Verkehrsverbund Tirol (www.vvt.at), there are sometimes a combination of two buses from Mittenwald to Innsbruck via Telfs Sagl or Seefeld, but I can't find them for every day. Mostly the connection is by train and probably faster that way.
I don't know of a bus (but there could be one, can't say definitively), but I have taken the train from Innsbruck to Garmisch-Partenkirchen. I think it would very likely be faster than a bus, anyways. The train I was on was one of the least nice trains I've been on traveling through Europe, but it was fine, no problems, and provided nice views along the way.
In general, buses in Germany tend to complement trains (i.e., go from a town with train service out to surrounding towns without trains) rather than compete with them (i.e., connect the same places). When they do connect the same places, as you see here with Garmisch to Innsbruck, it's often necessary to piece together a series of buses to go where the train goes directly.
Thank you all so much, I have looked at the train and it is probably our fastest route, I am looking at coming from Venice into Innsbruck and on to Garmisch spending the night and going up the zuspitze the following morn.