I'll be flying into Frankfurt this Saturday for the third time, I have never spent one minute in the city itself. Unless staying in a city, I always drive so can't say too much regarding trains. As noted, you will have the advantage of free public transport if you base in the Black Forest for a while. I'd skip Frankfurt and head to Gengenbach, a very attractive town with nice hiking close by. From there you can visit Baden-Baden for a day - I would but it receives mixed reviews here , Strasbourg for a day, some of the spa towns to the east (never been to any but they look pretty enough), Wissembourg if you have a car. I believe you can make a very nice and long day out of scenic train rides to Freiburg and back.
If you have a car, some places worth considering on your way down to the southern part of Alsace are Obernai, Monte St Odile, Andlau, Dambach-la Ville (not a tourist village, has a splendid church), Haut-Koenigsbourg. We based in Riquewihr with a car, from here we visited the other villages by car but took the bus into Colmar. Lots of nice, fairly easy walking. The Alsace villages are flush with flower boxes in season so I doubt you'll see any of that.
On the way to the Lake Constance area, if that's where you decide to go, there are the pretty villages of Staufen and Loffingen. Good hiking n the Leutasch Gorge near Loffingen. Lake Constance may be as dour as the alps that time of year. Meersburg is a great spot, Uberlingen close by, Sigmaringen worth a visit for the castle and the town itself, you can catch Lindau on the way to your alps base (lots of pleasant stops on the way).
I would rate Oberammergau as a practical base for visiting the places you mention ( I believe you mean Mittenwald). If the forecasted weather changes your mind before you leave, give some consideration to Franconia in northern Bavaria - anywhere from Wurzburg to Bamberg up to Coburg. Excellent hiking, lots of interesting sites and attractive villages/towns.
Munich worth three full days. I've only reached 52% of my 'maximum length' but I'm gassed out.