Sounds like the makings of a great trip. Seeing as you seem to have ideas around Grindelwald (don't forget the N face of the Eiger has a great singletrack underneath it- just don't go when there are loads of walkers as I think there may be a voluntary restriction on bikes there though that was a while back) I'll try and give you some ideas about the east.
There is loads of good riding around and between Lenzerheide-Valbella-Arosa that can be lift and train (don't overlook how good the sbb is for this) assisted. This is fairly well known about and signed etc. Just to the west, Flims and Laax are also good with some more DH/bikepark oriented stuff, and to the West Kloesters and Davos have lifts and trains that will help you get the most out of the riding. Seeing as you're up this way the riding up the valleys to Vals (famous for Zumthor's baths) and Vrin is pretty quiet and fun, as are some of the places around Savognin and Berguen (though not ridden as much here). Although some areas are reasonably "developed" for tourism and riding- it is the alps after all- there are some really fantastic little side valleys and passes all around the Buendner area that are great for exploring. Get a copy of the relevant singletrail maps and follow them- for the best stuff stick to the black dotted lines.
Between there and Tessin, I'd want to check out the Engadin areas more before heading down through Bellinzona- some good steep technical riding along with the 'classic' Monte Tamaro nearby, and I imagine the stuff around the lakes down there will come with a nice Italian flavour- similar to Lago di Garda etc maybe.
My advice would be to get familiar with postbuses and trains and try to use them to link together routes you really fancy doing- some of the best things I've done around there involved heading out alone over passes onto stupidly long singletrack descents to remote summer pastures, climbing back out into another valley and a tiny station (where you press a buzzer if you want to get on the next train), getting loaded into open goods vans and riding it out to Chur with some via ferrata soloing mit meinem velo into the bargain- happy days. With a car you are a bit restricted to the roads sometimes. The singletrails maps are very useful if expensive too.
For inspiration and info on specific routes I'm not too sure (especially in English), but try some of the following perhaps...
swiss tourism site, ride (the (german) swiss bike mag), some of the german mags- they often have alpine routes in Switzerland, maybe this- not used it before but maybe handy, and there is one other great site that I can't remember with great reports and images to get you started- possibly this although the site looks to have been redesigned.
Have fun.