I took down my old rain gutters and purchased new ones 10' each. How do I join the gutters together to make a 24' long section?

Go back to were you purchased your 10' lenghts, they sell couplers and end caps (left & right ends) I prefer to use pop rivets at the tops and gutter sealent to attach them.

Make sure you use a lot of sealant. You may be surprised how inexpensive having gutters done for you.
The company I use for say your 24' would be maybe $85 if that. And there's no seams to keep from leaking. And they will leak, the sealant don't last forever.
Do make sure when you do seal it that it have a full day or so to cure before any rain comes