As you might expect, the work you might do as a mathematician can be quite varied.
Most mathematicians engage in research. Broadly construed, "research" refers to the process of attempting to make progress in solving problems that currently have no solution. There are countless unsolved problems in mathematics (look up "millennium problems" for a handful of famous examples), and it's often the job of the mathematician to learn more about these problems. This involves extended study into mathematical topics, usually beginning at the college level (such as calculus, linear algebra, real and complex analysis, abstract analysis, probability theory, and many others). This is often followed by graduate studies in mathematics, which usually involve a combination of additional study into broad areas (like the ones above), plus focused research into a more narrow set of topics (based on the interests you've developed until that point).
Generally speaking, a research endeavor begins with a question. For example, "what is the shortest path between two points, assuming the path is confined to a plane?" One often starts by trying to gain some intuition for the question being asked. This might involve drawing a picture or writing down and quickly evaluating some first guesses at a solution. If you're lucky, you hit upon the correct solution, and you're able to prove it's correct. Otherwise, you might start cycling through a process that involves thinking about the problem, reading what others have written about the problem, and updating your guess at a solution and/or its proof. If you're lucky, the problem has already been solved. You might use this work going forward (always giving credit to the original authors), or use it to inform your own, new method of solving the problem. Once you've made a concrete step of progress, your job is to document how you did this, and share that work with others; your peers will review your work and ensure it uses sound methodology; your work is then generally published for the benefit of the broader scientific community.
Broadly speaking, there are two "types" of math -- pure and applied. Pure mathematicians tend to work on mathematical problems for the sake of mathematical progress; they find mathematical puzzles interesting enough to spend their lives learning about and trying to solve them. Their work may or may not have obvious or indirect relationships to real-world phenomena (though often such connections are made much later). Applied mathematicians tend to work on mathematical problems as a means of understanding or solving a problem in another domain (such as physics, data science, finance, medicine, etc.). Their techniques often require expertise not just in mathematics but also in the relevant external domain; working knowledge of programming languages are often useful for simulations and for achieving results usable in the external domain.
Your other questions are very good ones to ask. Research may be carried out individually or as part of a team; sometimes it is carried out as part of a geographically distributed team (ie, you might work with colleagues across the country or across the world). This really depends on the details of your particular situation. Most often, you'll do research as part of your graduate studies, and follow that group for at least a few years. You'll likely build connections from there and make decisions about whether to branch out.
One advantage of this field is that it is often possible to make progress with pen, paper, and computer (ie, no need for expensive laboratory equipment and lots of space or power to run it). This makes your expertise flexible and portable.
A mathematician might also choose to teach (often as part of the same position that pays him or her to do research). But I presume you've already seen what math teachers do, and so I've tried to focus here on the details of what you likely haven't seen a mathematician doing.
To sum up: mathematics can be thought of as the formalized study of patterns, with the aim of generating insight into a wide variety of phenomena (mathematical and otherwise). It comes complete with its own language, which happens to be the most precise and useful way to describe a staggering array of puzzles. Studying mathematics requires copious amounts of both time and work, but it is, in its own right, a fascinating and rewarding endeavor.
If you have any additional questions about being a mathematician, please continue asking -- curiosity is a critical characteristic of any good mathematician!
Source: I am an applied mathematician.