How do I recognize the difference between the different class carts (1st class and 2nd class) on German Trains: ICE and Regional Trains?

Yes, as indicated above it is normally very clearly marked on the outside.  The last thing the train companies want is people getting on the wring carriage and having to drag all their luggage through the train to the correct carriage.

In practice there are normally 2 ways to tell which is 1st and which is 2nd.

1. On the door, close to the open/close button there will be a big “1” or “2”.  Possibly a “1/2” if the carriage contains both, although this is less common on long distance trains.
2. The roof-line of the carriage will have a yellow stripe for 1st class, so you can see from a distance as the train pulls into the platform which ones are 1st and which are 2nd.  You can see the yellow stripe on the front 2 carriages in this image below.

And finally, often there will be train layout posters on the platform billboards showing the normal formation of trains showing car numbers, platform location (look for big “A”, “B”, “C”, etc . . . signs on the platform), and car class.

In reality, there’s not much to it and once you’ve got your first train you should have the idea.  Also note that there’s no problem with walking through a 1st class carriage with a 2nd class ticket.  You can’t take a seat, but you can pass through no problem.  Quite often you have to do this since the bar/bistro/restaurant car is normally beside the 1st class cars.