Mathematics without coordinate systems

Suppose that

1. I understand mathematics to the level of e.g. Rudin's "Principles of mathematical analysis"
   

2. I want to completely expel "coordinate systems" and the resulting "analytic geometry" from my mind, and rebuild my knowledge and expand it, always keeping in touch with geometric intuition.

The reason is, briefly, that I have always felt "coordinate systems" as a dive into a world of unnecessary projections, a sad departure from the joy of mathematics done in the style of Euclid and also in the style of Newton.

So my questions are:

a) Do you think the above is a naive feeling showing lack of mathematical maturity?

b) What general plan of study do you propose, that leads to mastery of geometry, algebra and analysis, at the level of a modern university degree in mathematics?

Of course I do have my own thoughts and plans, but I would also like to learn something from other people, more experienced and possibly like-minded.