If you're interested in surface meshing, the book Delaunay Mesh Generation by Cheng, Dey, and Shewchuk is very good. Surface meshing is a pretty involved topic and I couldn't attempt to summarize it here. The constrained Delaunay triangulation and Voronoi diagram are especially important.
You also mentioned being able to find seed points that are on the surface. Part of this will involve being able to solve nonlinear equations, for which you'll want to use Newton's method. If you're dealing with algebraic surfaces, Smale's $\alpha$-theory can give you convergence guarantees that aren't available in general. My old officemate gave a talk (slides) that I think makes a great introduction. You can also take the sum of the squares of all the equations defining the surface and think of it as an optimization problem, but in that case Smale's theory applies too.
Assuming you do have some point on the surface, another important operation is to move that point following the flow of some vector field. You might also want to be able to compute geodesics. This amounts to solving a differential-algebraic system. Geometric Numerical Integration by Hairer, Lubich and Wanner has a lot of information about ODEs on manifolds.
For a real application, you can check out the manifold class in the deal.II library. deal.II uses manifolds to solve the problem of refining initially coarse meshes on surfaces. If you try to refine a mesh of, say, an annulus, a naive implementation would just put more points on the boundary segments without adjusting the curvature. With deal.II, you can say that a certain part of the boundary of a mesh is actually some manifold, and the refined mesh will adapt to the real surface much better.
Finally, you might find a lot of the engineering literature on boundary representation or constructive solid geometry modelling to be inspirational. Sounds like a cool project and good luck! Do post a link on here if you make any of it open source.