I'd skip Tegernsee, it's pretty, but you may have other things you really want to do. Also - I think Italy's a bit far for a day trip from Mittenwald.
We were just down in this area this summer visiting my brother in law. We're heading down again in a few weeks when my parents fly over from the US. I say skip the Zugspitze and do the mountain right next to it. It's cheaper and more scenic. You can go up the Kreuzeck on a very modern cable car then take an easy walk to the next cable car that takes you to the top of the Alpspitze (or you can hike that middle section, not too bad at all). Walk around up there then take it's own cable car straight down to the bottom. You end up making a triangle up and across the mountains while getting a fantastic view of the Zugspitze since you're right next to it. Much more fun.(IMO)
Also-if you go to Eibsee with your car you'll have to pay a parking fee. You can walk there from the furthest Zugpsitze connection, but you should check on whether or not it's ok to leave your car there so long.
Right outside of town in Mittenwald is a really cool looking gorge hike. The TI in the middle of town can give you maps, as likely can your lodging reps. There are suspension bridges, etc. and it's supposed to be kid-friendly. We ran out of time this summer, but are going to try to do that when we head down again in a few weeks.
Also-we had Frommer's 25 Best Driving Tours in Germany with us, along with our GPS, and it was fantastic, lots of fun, good ideas, and Mittenwald is where 2-3 tours overlap. Just heading north from Mittenwald towards Munich (~10 min north of town?) you can take a private road where you pay 3 Euro, and drive alongside the river-very pretty, places to stop on the way; we give it 3 thumbs up. Worth finding.
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