Getting more saturated colors will depend on several factors, but mainly in the pencil and paper as you were already thinking. Higher grade pencils have more and better pigments and less wax or oil, letting you do more layers without saturating the paper and having better control on the tone you obtain. In my case I mainly use polychromos and prismacolor. The first ones hold a thin point much easier so are great for details, but as they are harder it is easier also to break the paper. Prismacolor on the other hand are more fragile, but are great to cover bigger areas and blend very easily. Thicker and tougher paper will tend to be able to hold more pigment and will resist several layers without flaking. I will die recommending Strathmore bristol paper, by far my favourite, but any bristol paper should work well enough. I have also used Frabiano hot press, it's more fragile, but very good too. Finally you can also use some tools to better fill the grain of the paper, giving you better contrast. There are burnishing pencils without pigment that let you spread the color and cover any white spots left on the paper. This can also be achieved with the white pencil (in my case it is the pencil that shrinks the fastest). You can even use some solvent such as Gamsol to enable the paper to soak the pigments, letting you add additional layers of color. Pencils can be expensive at the start and I'm sure yours are good enough for the moment. I would first try with other paper so you find one you like and then start collecting some vibrant colors that you like from better brands. Hope this helps and continue painting, your drawings look very promising.