I have something to add to this. All through high school I fought with my high school math department to be excused from high school math. For example, I said it was stupid to be taking high school trigonometry while also studying differential equations at Columbia. My tenth-grade trigonometry teacher said that just because I was studying differential equations at Columbia, did not mean I did not also have something to learn from tenth-grade trigonometry.
Looking back on it now, with the wisdom that comes with age, I can see that I was right and he was wrong. I had nothing to learn from tenth-grade trigonometry.
My ninth-grade math teacher asked why I was always “trying to do an end-run around life”. I suppose that to this ninth-grade math teacher, your ninth-grade math class is one of life's highlights, one of those fleeting moments of youth that one must stop to savor lest it slip away all too quickly.
But I think this story proves that I wasn't just trying to rush ahead for the sake of rushing ahead. When I was offered the chance to do two semesters of real analysis a second time, instead of rushing ahead to the next thing, I didn't try to skip it. I didn't even argue.
Those high school people were wrong. So, so wrong.
May they all burn in hell.
[ This is another resurrection of a deleted Math Stack Exchange post. There's nothing really wrong with it, except that I feel like it's not of general interest. ]
When I first entered university, shortly before classes began, I met with an professor whose task was to advise me on which classes to take in my first semester. After hearing me describe my background, which included passing the college-credit calculus exam at age fifteen, he suggested that I take real analysis.
“But I took that already,” I protested. “I had a two-semester course in real analysis at Columbia University last year. We used the little blue Rudin book. I got A’s.”
The professor said to me that analysis was a deep enough and rich enough subject that I would not be wasting my time to take it again, and that I would not be bored. I thought about this a little bit, and then I agreed that he was probably right. So I took the analysis course again. We used the same textbook, but I was not bored, and it was not a waste of time. It was an extremely good use of time; I have never regretted it.
So that's my answer about which topics of real analysis should be studied if you have already done calculus: all of them. You will not be bored, and it will not be a waste of time, because the answer is the same even if you have already taken real analysis.