Nobody told me that self teaching could be so damaging...
There are many things you can do.
If you do understand the homework then it won't take you very long to write down the solutions. Go ahead and do it. Try to do it elegantly.
Look for the subtleties, I tell my undergraduate students that they will really understand the material of course X after they teach the course.
In your text there may be more advanced problems, try them or try looking in another book/source.
If you feel that you can handle more advanced material, try getting permission to sit in a course that interests you, try doing the work from that.
It is good advice to talk to the chairperson or a faculty member. Do make sure that your tests/homework are coming back as correct. If they aren't, it is still a good idea to talk to someone (it always is!) but then you might want to present your situation differently.
Eventually, all Mathematicians are self-teaching. Hopefully, on the course from Elementary School to Graduate School, we get weaned off learning from our teachers and more on learning for ourselves.
There are people who are comfortable within the Academy and many people who are suspicious of it.
Oh, yes. I’m very proud of not having a Ph.D. I think the Ph.D. system is an abomination. It was invented as a system for educating German professors in the 19th century, and it works well under those conditions. It’s good for a very small number of people who are going to spend their lives being professors. But it has become now a kind of union card that you have to have in order to have a job, whether it’s being a professor or other things, and it’s quite inappropriate for that. It forces people to waste years and years of their lives sort of pretending to do research for which they’re not at all well-suited. In the end, they have this piece of paper which says they’re qualified, but it really doesn’t mean anything. The Ph.D. takes far too long and discourages women from becoming scientists, which I consider a great tragedy. So I have opposed it all my life without any success at all. . . -- Freeman Dyson
Any discussion of (successful) autodidacts has to include Srinivasa Ramanujan. From what I understand he read his textbooks very carefully and built from that. His isolation from the Mathematical community meant although he reproduced old results, he did so independently and offered a new substantial point of view.
It was in the Town High School that Ramanujan came across a mathematics book by G S Carr called Synopsis of elementary Results in Pure Mathematics. This book, with its very concise style, allowed Ramanujan to teach himself mathematics, but the style of the book was to have a rather unfortunate effect on the way Ramanujan was later to write down mathematics since it provided the only model that he had of written mathematical arguments. The book contained theorems, formulae and short proofs... The book, published in 1856, was of course well out of date by the time Ramanujan used it.
His acceptance into the mathematical community was not instant, but gradually with his correspondence to mathematician GH Hardy at Cambridge. From his first letter:
I have had no university education but I have undergone the ordinary school course. After leaving school I have been employing the spare time at my disposal to work at mathematics. I have not trodden through the conventional regular course which is followed in a university course, but I am striking out a new path for myself. I have made a special investigation of divergent series in general and the results I get are termed by the local mathematicians as 'startling'.
John Kingman known for the his paintbox process for Random Paritions (see also here on paintboxes). He only has a Master's Degree and he currently advises students at Oxford.
Why not both? My Suggestions;
For each course you do, try and find a higher level book to complement it.
Many first and second-year course introduce concepts which you will generalize later in your studies. Find an excellent book that covers the course material in a more general setting. For example, if you're studying linear algebra, maybe try a book on functional analysis, for multivariate calculus, try a book on differential geometry. Do your homework and tests, but try and see how your course content relates to the bigger picture.
Learn to program.
It doesn't matter what language you start in (i prefer object orientated languages), but learn the fundamentals of programming and learn how to use it to solve mathematical problems. Use it to aid visualization of assignment problems. Check the software output against your intuition and try to describe any significant numerical errors mathematically.
I would argue that programming is an invaluable tool for research and an extremely important skill for math graduates looking for work outside academia.
Learn latex, use it for all your assignments.
I found that learning to typeset my work forced me to think hard about the communication of a mathematical argument; to identify the important steps and to omit the trivial stuff. But most importantly, it pushed me out of the high-school (early undergrad) idea that maths is a sequence of equations, and English only appears at the start and the end of a piece.
Focus on developing understanding and intuition
This has been mentioned in other answers and I would say that my other suggestions all lead towards this. Make sure you're learning the ideas not just the procedure.
Hope this is useful.