How much are you taking to Europe? The one great thing about backpacks is that they fit in lockers at the train stations in Germany. If you're just stopping for a few hours, leave your bag, go in and tour, and then come back to the train station to continue on to where ever city you're spending the night.

The train stations in a lot of cities you'll be visiting have locations near the city center, or easy connections to local transit, so you're really not going to be hauling that backpack on your back for any long distance.

I was over again just this past February and did stays in Berlin, Leipzig, Munich, Salzburg and Frankfurt with individual train tickets bought about 6-8 weeks out, all at a discount. I was with friends and we all had rollaboards (had to give up the backpack after 20 years) and hotels near train stations or transit. With some planning you should be able to work around any concern about traveling with a backpack.

You asked about Prague. I've done it as round trip from Berlin. You can buy those tickets on line at the German Rail site.

Don't limit yourself to just hostels. Look at Motel One which is a budget chain with locations around Germany. For two people, they could come out close to the price of a hostel (typically 59-69 euros in my stays).