It depends on your aptitude.
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if you are in honors math you take geometry your freshman year, and if not your sophmore year. the order goes per-algebra, algebra 1, geometry, algebra 2, pre calculas, and then calculas (calculas is only taken for AP classes, for seniors that want to take it. Calculas counts for a college credit.)
If you were somehow able to take algebra 2 without geometry then you most certainly can. However, that is a difficult feat. Good luck with that, if you try it.
There are more than three types, although 2 main types are Euclidean and Taxicab Geometry
solid geometry deals with 3 dimensional figures while plane geometry deals with 2 dimensional.
The 2 types of non-Euclidean geometries are hyperbolic geometry and ellptic geometry.