Is it cruel to take one dog somewhere while leaving the other at home?

Dogs do NOT think the same way humans do. If you are taking your puppy somewhere that you KNOW your 1-year-old dog doesn't enjoy going, your older dog isn't going to be mad at you. That being said...your 2 dogs may benefit from being in the same places together so that they can learn acceptable behavior together. Your puppy doesn't have many experiences in the "real world" and your older dog does. SO, your puppy might learn the appropriate behavior you expect if he/she is watching the older dog in the same situation.

I think your reasoning is sound, that there will be some places where you need to take the puppy and you don’t need to subject your older dog to accompanying you.

For example at the vet, there will be pets around who are sick including canine influenza, which is currently on the rise. Your older dog’s immune system may not be as robust plus most dogs I know do not enjoy being at the vets. For socialization and training practice, you can be more dedicated and focus more on your puppy becoming a good canine citizen if 100% of your attention is on puppy, rather than having to split it between the two dogs. When puppy’s training&socialization is more advanced, the older dog may fit better into that. To help achieve balance you, might opt to take your older dog some places without puppy. And of course give each of them attention at home, which is probably what they crave most of all.

Yes it is. And anyone saying they don t think like that is dead wrong. When you come back, your dogs will smell him all over and know exactly where you took him. I have had my dogsister beat up the returning dog on more then one occassion. They all like to go, Don t let anyone tell you otherwise