Same rocket, different payload?

Here's how I'd do it:

1. Design Apollo by itself.
2. Save it as a new craft named "Apollo".
3. Design Saturn as the rocket underneath Apollo.
4. Save this new craft as "Apollo+Saturn".
5. Disconnect Saturn from Apollo and save it to the Subassembly section of the parts list in the VAB/SPH as "Saturn". This will remove the parts from Saturn from the craft.
6. Save the current craft (which is just Apollo) as "Apollo+Atlas". (This ensures you don't accidentally overwrite anything with future saves.
7. Design Atlas as the rocket underneath Apollo.
8. Save the updates to the craft, again to "Apollo+Atlas".
9. Disconnect Atlas from Apollo and save it to the Subassembly section as "Atlas".
10. Create a new, empty craft.
11. Design Mercury by itself.
12. Save it as a new craft named "Mercury".
13. Retrieve Saturn from the subassembly list and attach it to Mercury
14. Save this new craft as "Mercury+Saturn"
15. Reload "Mercury" (or just disconnect Saturn).
16. Retrieve Atlas from the subassembly list and attach it to Mercury
17. Save this new craft as "Mercury+Atlas".

Now you've got two payloads as main craft, two lifters as subassemblies, and all four combinations thereof.

You can save your different launchers and payload separately and then save them as subassemblies.

When you are finished designing a component, switch the part menu to "Advanced" mode (arrow in the upper left corner) and go to "subassemblies", drag the whole craft design to the "subassemblies drop zone" in the lower part of the subassemblies menu and give it a proper name. When you get an error message "THE SELECTED PART IS NOT ATTACHABLE", change the root node to some part which can be attached radially or has a free attachment point at the top or bottom.

You can then access that saved collection of parts from any new rocket design. Hint: Start your combined design with the decoupler you intend to use to link the two components.

You may also want to look into SelectRoot, as it can make designing, editing, and combining parts easier.

In KSP, the "craft" is the root part. All other parts are attached to either the root part, or parts that are attached to this root part. (FYI, this is known as a "tree" structure in computer science.) You can only save, load, and launch whole crafts in the VAB/SPH.

You can disconnect any part that's not a root part and save it to the Subassemblies tab. That part, and any part attached to it, will be saved as its own collection of parts. You can then retrieve those subassemblies and add them to other craft.

With careful planning, you can create nearly any combination of craft & assembly you like. I've used a dummy control module as the root part of a craft to specifically design a subassembly for it's own sake.

However, it's kind of a pain to work that way. With SelectRoot, you can change which part is the root part in any craft. Then you can attach/detach/save subassemblies as you like. With this, it becomes fairly easy to design all sorts of modular spacecraft.