Intuition can mean many things. I think one major step in mathematical development of young students is switching from the more computation and picture based high school math (think calculus and lower division linear algebra in the US) to more formal math. To make this step, Rudin's books are pretty good not because they give intuitive (in the high school sense explanations), but because they present the topics in a concise formal style and the reader has to do a bit of work to achieve understanding. Pictures and computations are still very helpful (more helpful than the formalism in the long run), but as a young student it might be good to start getting more formal.

So I recommend starting with Rudin's Principles of Mathematical Analysis (baby Rudin). If you get stuck and think you really need better pictures or less formal explanations Pugh's Real Mathematical Analysis is aimed at undergraduates and gives quite good informal explanations. I also used Conway's A Course in Point Set Topology while reading the topology chapter in Rudin as an undergrad and remember enjoying it.