This was one of the aspects of algebra that I enjoyed the most while first learning it. This is the way I would develop the subject:
Introduction to exact sequences with an emphasis on short exact sequences. We can use these to illustrate how the properties of any module in such a sequence are related to those of others. Relevant here would be discussions of theorems such as, length of a module is additive, a module has ACC/DCC if and only if a submodule and quotient modulo the submodule have ACC/DCC. We can also develop direct sums via short exact sequences, the short five lemma, the snake lemma here.
Projective modules: Given a short exact sequence, $0\to L\to M\to N\to 0$, of $R$-modules, and some other $R$-module $T$ and a homomorphism $T\to L$ there exists a homomorphism $T\to M$ via composition. We can ask the opposite question. When does giving a homomorphism $T\to M$ give a homomorphism $T\to L$. This can be used to motivate $Hom_R(T,-)$. Then we can go on to show this functor is left exact. Then we define projective modules as those which make this functor exact.
At this point I would introduce Free modules and motivating them via vector spaces as you suggested. I would also discuss free resolutions since this is a very elegant machinery. Then, I would show that projective modules are just direct summands of free modules.
Injective modules: A similar development as for projective modules, this time via the functor $Hom_R(-,T)$. I would also develop the characterizations of injective modules via Baer's theorem and divisible modules.
Flat modules: I shall then introduce the functors $T\otimes -$ and $- \otimes T$ and show these are right exact. Then define flat modules as those which make the above functors exact. I shall also discuss that projective modules are flat.
To tie all of this together, I shall then discuss the adjointness of Hom and $\otimes$ followed by Homological algebra if that is part of the course.