I also strugged with analysis. I always felt like epsilon-delta proofs involved pulling some absolutely strange value for delta out of your ass that happens to work out in the end, and I never developed an intuition for that. Same with integrating by parts, oh it just works out so nicely if you rewrite u in this totally obtuse way.
The tricky thing with analysis that I don't think many professors are good at conveying is that the ordering of statements in the proof isn't the same as the ordering of steps the proof writer performs to come up with the proof.
Basically you sort of write the broad strokes of the proof up front, leaving the right hand sides of statements like "Choose epsilon such that epsilon = __" blank. Then you do a bunch of scratch work to figure out what epsilon needs to be so that your proof works out in the end.
Another challenge with analysis is that inequalities are central, so fluency in their manipulation is absolutely critical. And naturally most students aren't fluent with them by the time they take analysis, so they get bulldozed by Baby Rudin and learn to hate a pretty cool (and useful) branch of math
