"At the age of 10 I started swimming" or "At the age of 10, I started swimming" Which one is correct?
No, you don't need a comma there. The sentence is perfectly clear without.
You would need one in the following sentence, because you would naturally pause after the word "ten" if you were speaking it:
At the age of ten pupils were encouraged to start swimming.
At the age of ten, pupils were encouraged to start swimming.
And this sentence needs two to show that "when we moved to Southend" is parenthetical:
At the age of ten, when we moved to Southend, I started swimming.
Reading a sentence out loud, as if you were speaking to someone, can be helpful when deciding whether a comma is needed or not.
Many guides will tell you to use a comma after an introductory phrase or clause, but nobody would mind if you omitted one in your example. The meaning is perfectly clear. (https://iue.edu/hss/writingcenter/documents/Commas.pdf)