I look young for my age. I look younger than other people who are the same age as me.

These two sentences are redundant. You really only need to say: "l look younger than others my age"

Hey Sean, The first sentence is okay, but the second would be more natural with an "of" instead of a "at". "at my age" is more appropriate to use in a context of a conversation between a younger and an older person, in a context where the older person says to the younger one "when you will be at my age, you will understand what love means" Also, in a different sentence would be natural to use no connection word at all whatsoever, so you can say "people my age are more socially active than I am", whereas in the sentence you wrote, an "of" is appropriate.