Subject-Verb Agreement + Prepositional Phrase Argument

Solution 1:

I'm learning. (base)
I'm learning how to do things. (adjective phrase)
I'm learning how to do things in a way. (adjective phrase)
I'm learning how to do things in a way that serves me. (adjective clause)

"a way that serves me." is the important part of the sentence.
"that" is a relative pronoun.
the antecedent for "that" is the adjacent noun, "a way".
"a way" is a singular noun.

a way serves me. // correct
a way serve me.

NOTE
If you want to try and say that "that serves me" modifies "things", then you would have:
I'm learning how to do things that serve me. // ok. "things" is plural. but wait...
I'm learning how to do things that serve me in a way. // grammatically correct. The antecedent for "that" is now plural, "things". But the dangling prepositional phrase, "in a way", does not make logical sense.