As for differential geometry, the minimum prerequisites are linear algebra and a rigorous calc III course. I suggest you go through something like "calculus on manifolds" by Spivak.
Then you can read books like Do Carmo's "Curves and surfaces".
Although not strictly necessary, I would suggest learning topology as well. Thinks are going to make much more sense with some topology knowledge. Either way, if you want to read advanced books on differential geometry, then you will need topology anyway.
As for algebraic geometry, you will need quite a solid knowledge of abstract and commutative algebra. You should be very comfortable with things like ideals, prime ideals, modules, etc.
You don't need to be an expert in commutative algebra however, it is possible to learn the commutative algebra while you are reading algebraic geometry. Just make sure you know the basics.
A basic knowledge of topology is handy as well.
A good book is Harris "Algebraic geometry: a first course".
As for homological algebra, it is a subset of abstract algebra. But you don't need to know homological algebra right away. Besides, it is going to be very weird and unmotivating to you. Of course, one you go deeper into differential geometry, algebraic geometry or algebraic topology, then homological algebra will show up. But by then, it might be more motivated to you.
Another excellent first book in algebraic geometry is Miranda's "algebraic curves and Riemann surfaces". It is totally different than the Harris book, but it is really great in motivating all the concepts. To handle it, you will need to know complex analysis, differential geometry and some abstract algebra.
