People have done this, but most don't succeed. With Differential Geometry, the main backstop is a willingness to actually do computations. I tried learning some stuff about manifolds on my own as an undergrad, but the motivation for me to actually sit and compute a transition function or a flow or whatever was at an all time low, especially when I knew how it "should" work. If you end up doing this, don't give in to that temptation. Knowing how it works is the easy part; actually doing the calculation to verify your intuition is the defining barrier between knowing and not knowing differential geometry.
Algebraic Geometry, on the other hand is not something I'd recommend trying to self study, at least without someone who knows the subject regularly available to answer your questions. There is simply just too much necessary material and necessary machinery, all of which will seem very foreign to you unless you've taken a number of very dense algebra courses. The prerequisite courses required are also quite a barrier to entry. In my opinion, the bare minimum to take algebraic geometry is a graduate course in commutative algebra (at the level of Atiyah Macdonald). To actually get something out of the course, I also feel that it's important to see the concepts used in a simpler, easier context first. For instance, studying curves/surfaces over the complex numbers only makes sense if you've seen some complex analysis. The language of sheaves/schemes is hard to parse unless you've seen some stuff about manifolds (R\^n is to a manifold as an affine scheme is to a scheme, and a manifold's atlas is like a scheme's structure sheaf). Sheaf cohomology is going to make no sense unless you've seen simplicial or DeRham cohomology from either differential or algebraic topology.
If you've taken (and done well in) all the courses I just mentioned, then \*maybe\* you can self study algebraic geometry. Otherwise, without a professor/expert to help you with the intuition, you will (probably) be dead lost.