Developing Mathematic Intuition

One of my teachers always told me "don't know definitions, don't know math." At the time I was pretty annoyed, but he was completely right. The only way to learn math is to have the fundamentals down cold. This involves both a rigorous side, (memorizing them is a good start) and an intuitive side. So at an entry level, I strongly recommend spending a long time with the definitions. Theorems are nice and can help you understand the relationship between the definitions. But as far as Intuition goes, don't dive into the mechanics of the theorems too early.

Some big ones from calculus are limit, Taylor series, integral, derivative/differentiable, open/closed, even/odd, and continuous. If you know those you can probably talk to anyone about calculus.

The only way to build your intuitive understanding is to fail. Getting it wrong is the first step to getting it not totally wrong. That means trying a lot. Do your homework carefully. Try to ask follow up questions. A good curriculum can help reduce the amount of time it takes, you'll have to be patient no matter what. Do examples. Do hard examples. Do more examples. Do counter examples. Do not just settle for "well, $0$ satisfies the equation so it's probably fine." We've all done that, but it's bad practice.

You know you're on the right track when you can see why a definition was picked the way it was. That is the real heart of intuition for definitions. For example, why should the coefficients for Taylor series look like they are? What properties do we even want from a taylor's series? Well, polynomials are awesome and simple. So let's use polynomials to approximate stuff. Ok... but how can we pick good approximations? It turns out it has something to do with making the $n^{\text{th}}$ derivative have the right value. It's worth understand how that works.

It sounds like you're on the right track. Half the battle is wanting to do it. The other half is work.

Also, this site is a good resource. Learning to ask good questions here will be super helpful for you.