Math is a field that can be frustratingly self-serve in terms of outcomes after graduation because the career paths are somewhat poorly defined. I’d broadly characterize the different paths as academic, education, financial, and engineering. In terms of academic rigor I’d say generally education < finance <= engineering < academic. In terms of money I’d say academic < education < finance < engineering. It was an observation of mine (as a graduate with a double major in math and engineering) that despite the fact that math was as difficult a major as engineering, engineers tended to make more money because of supply and demand (similar story for engineers vs doctors for what it’s worth). Also, for math your outcomes depend very strongly on how you structured your own learning - it isn’t standardized the way engineering is.
On proofs: they exist, they might be hard if you’re not used to them or not good with logic. Honestly I wouldn’t worry about it - if you have a strong natural understanding of math and mathematical principles from calculus, then mathematical logic is pretty intuitive. Most people who struggle with proofs either memorized their way through calculus or weren’t good at it in the first place. To be honest I think you should know by calculus if math in general is for you.
On the different disciplines:
Academic is the kind of math that leads to graduate school and a doctorate program, then a job in math research. You probably want to do mathematics research for this path, focusing on the more theoretical side of math. You also want to learn all or at least most of the traditional mathematical fields, the kind that are associated with the qualifying exams. A decent list of them is given by UCLA: http://www.math.ucla.edu/grad/handbook/quals
These topics are very tough if you want to learn them in any depth, and to put it very simply, I will say that if the academic path seems like it might not appeal to you, then you probably won’t change your mind. Adjusting for intelligence, it’s one of the lowest paying careers there is because your only employer is academia.
Education is a path to grade school and community college teaching. Academically it’s probably the simplest - you won’t be teaching past calculus, or DiffEq/LinAlg if you’re particularly ambitious. You will probably need to take a standard math sequence up to analysis, including geometry, and some courses on teaching. Generally it doesn’t pay that great, but offers good job security and good benefits. If you’re smart enough to do a PhD in math at a top school then you could probably find one of the few high-paying teaching jobs as well.
Finance is the Wall Street/actuary path. The actuarial exams as previously mentioned are strongly rooted in probability, statistics, and economics. Also very useful is a working knowledge of programming (computer science classes and computation classes) and classes for dealing with data (linear algebra and numerical linear algebra). It pays pretty well, though it’s not particularly easy. Financial institutions and insurance companies are the standard employers here.
Engineering is generally focused on mathematical solutions to engineering problems. It can involve programming and computation (computer science and classes like numerical analysis and scientific computing), solutions to physics problems within engineering (differential equations and numerical analysis), and advanced quantitative/computational skills in general (linear algebra for data, probability and statistics, lots of different software skills, some engineering experience). If you play it right then you can manage an engineering job with the math degree, which pays as well as an engineering job does. Usually it’s an “applied math” job of some sort in an engineering capacity. Closely related is applied math work in a more academic “science” capacity, which would fall somewhere between “academic” and “engineering” in both rigor and financial considerations.
To put it simply, a math degree is what you make of it, and if you get one without a goal in mind then you will probably get nowhere. You should figure out what you want to do with it and pursue that goal (choosing schools and classes accordingly) to pursue one (or more) of the paths as above.