I think that it is good to start with basic divisibility. There are a lot of exercices/problems you can solve only using some basic techniques like prime number factorization, GCD and maybe Bézout’s theorem. After that, you could start learning some more advanced concepts - like quadratic residues, Euler’s totient function, Fermat’s Little Theorem, Wilson’s Theorem, the Primitive element theorem and start using these in order to solve some more advanced problems like diophantine equations. You can also learn some basic properties of multiplicative functions.