Geometry is (according to the Oxford English Dictionary, OED) 'the science which investigates the properties and relations of magnitudes in space, as lines, surfaces, and solids'. Geometry was known to ancient civilizations but a major early collator of ideas was Euclid of Megara. Of course, the object geometry starts with is a point, which is a dot shape that takes up no space, but a line is a type of shape made out of points stuck together. All other shapes are made from points and lines. Shapes include circles, triangles, squares, spheres, cubes, cylinders, cones, and countless others. It is important to learn about points and shapes because they make up everything. Actually, points make up everything, but shapes can make things in the mind. When something exists in the mind, it can sometimes be used in the real world.
Formal geometry is not just about space and shapes. It is about proving ideas about the shapes. In school such as grammar/high school as well as some colleges/universities, students have the option of learning formal Euclidean geometry. Formal and informal geometry require learning algebra first. In formal geometry, one takes an algebraic idea such as one about a point or type of line, and one uses reason & logic, equations, and steps, to prove ideas about shapes.
Geometry also includes trigonometry, which is (according to OED) 'that branch of mathematics which deals with the measurement of the sides and angles of triangles, particularly with certain functions of their angles or of angles in general (the sine, cosine, tangent, cotangent, secant, and cosecant,) and hence with these functions as applied to abstract quantities; thus including the theory of triangles, of angles, and of (elementary) singly periodic functions.' Geometry also includes analytic geometry, which may not be taught much anymore but has applications such as in theoretical computer science. In the advanced formal geometry students can learn for a Master's or Doctorate degree, they learn non-linear and non-Euclidean geometry. There are even shapes that exist in four, five, six, dimensions and so on to infinity. Linear shapes are polytopes, which include polygons, polyhedra, and in four dimensions, linear shapes are called polychora. The names for shapes in dimensions higher than that are not agreed upon, but in four dimensions, there are six regular polytopes, and in five dimensions and higher, there are three regular polytopes in each dimension. This does not include star polytopes, which are regular in other ways, and there are many other types. There are also higher-dimensional curved objects such as hyperspheres, etc., and geometry also studies different and higher-dimensional spaces such as Hilbert space, which includes infinite-dimensional space.
Mathematicians should study formal Euclidan geometry (and perhaps other types after studying analysis/calculus) and prove theorems.