Where does Euclidean and non-Euclidean geometry lead to? I'm thinking of whether I should self-study Euclidean, elliptic, and hyperbolic geometry but I don't know where these subjects lead to.
Are they prerequisites to algebraic geometry, differential geometry, projective geometry, etc.? How are they related to other subjects?
 A: The short answer is that in mathematics there are often profound (and surprising and gratifying) connections between different areas.  You might study topic X for its own intrinsic attractions, or you might get into it only because you're fascinated with another topic or problem Y that has connections to field X.
Beyond that, I'd suggest web searches on the connections between topics.  For example, the search "hyperbolic geometry and number theory" turned up lots of results, including

*

*What are the interesting applications of hyperbolic geometry?


*The Oracle of Arithmetic: At 28, Peter Scholze is uncovering deep connections between number theory and geometry.
From the latter:

In the middle of the 20th century, mathematicians discovered an
astonishing link between reciprocity laws and what seemed like an
entirely different subject: the “hyperbolic” geometry of patterns such
as M.C. Escher’s famous angel-devil tilings of a disk. This link
is a core part of the “Langlands program,” a collection of
interconnected conjectures and theorems about the relationship between
number theory, geometry and analysis. When these conjectures can be
proved, they are often enormously powerful: For instance, the proof of
Fermat’s Last Theorem boiled down to solving one small (but highly
nontrivial) section of the Langlands program.
Mathematicians have gradually become aware that the Langlands program
extends far beyond the hyperbolic disk; it can also be studied in
higher-dimensional hyperbolic spaces and a variety of other contexts.
Now, Scholze has shown how to extend the Langlands program to a wide
range of structures in “hyperbolic three-space” — a three-dimensional
analogue of the hyperbolic disk — and beyond. By constructing a
perfectoid version of hyperbolic three-space, Scholze has discovered
an entirely new suite of reciprocity laws.

and, speaking to whether you learn topic X first, or get led to it by topic/problem Y:

“I understood nothing, but it was really fascinating,” he said.
So Scholze worked backward, figuring out what he needed to learn to
make sense of the proof. “To this day, that’s to a large extent how I
learn,” he said. “I never really learned the basic things like linear
algebra, actually — I only assimilated it through learning some other
stuff.”

So by all means start studying geometry.  If you don't like it, don't worry:

*

*you may never need it

*the textbook you're reading is not the right one for you

*the time when you need it for topic/problem Y may be a better time than right now.

On the other hand, learning it now may make you better prepared to tackle topic Y when the time comes.
Some other interesting reads:

*

*Series, The Geometry of Markoff Numbers

*Stilwell, Geometry of Surfaces
