My best advice you any beginning student:
1) Learn as much as you can,as can, as broadly as you can:. Take as many hard core mathematics courses-especially proof-oriented ones-as you can. "As you can" is the most important part of this statement here,because mathematics takes a lot of time and discipline to learn.As a result,there are very real time constraints on what you can learn at one time in one semester.You also want to sample many diverse fields where mathematics has a role,which is just about everything. You want to be as well-rounded and flexible as possible in your base knowledge because that's what ultimately becomes your foundation as a researcher.
2) Know thyself and plan accordingly. It's a lot better to take only 3 hard core mathematics courses in one semester and get B's and above in them then to try and take 8 courses and barely squeak by. Maybe you're the type that can go days at a time without sleep running on caffiene and/or....other chemical assistance and fear without landing in a hospital and ace 9 courses.Even if you are of that priviledged few-it'll catch up with you sooner or later,trust me. I've buried a few friends that it caught up with just as they finished thier PHDs at Stanford or Oxford.It's better to do less and do it better for not only your grades,but your health. Quality not quantity. If you've got health problems and personal issues that will get in the way of a total commitment to your studies and undermine your performance-as was the case for me-you should consider taking a few semesters off to take care of that. Otherwise,you could be a very sorry student later. Trust me.
3) Grades Matter. I know,you're probably like,duh? But this really hits on your question about the comfort zone-because throwing yourself outside it to prove yourself a badass is risking your GPA. Graduate level mathematics is serious buisness. Indeed,a number of such classes will be Moore method type courses where you're basically on your own and will have to prove basically all results yourself and a good portion you'll be graded on. And if you do poorly-it could yank your GPA down destructively. Getting into as prestigious a graduate school as you can manage determines how successful you'll be early in your career and whether or not you'll be banished to obscurity in some community college. My point is you don't want to risk that being a badass,it's not worth it.I would strongly advise you audit such courses at first.That way,you learn the high-level material and your performance doesn't affect your grades.Once you've gotten your feet wet-by all means,try and take ONE graduate course.Then depending on how it goes-take more.
4) Talk to People. Develop good bonds with willing professors and graduate students.This way,you learn about the field from many other people's perspectives as well as learning about thier experiences. It's also good to hang out with people who enjoy what you do!
5) Start Reading Journals. This is probably something you're not really going to be able to do until your last year before graduate school simply because you don't have enough background. But there are a few journals that are written at a low enough level for undergraduates to understand-like the American Mathematical Monthly. But you definitely should try and get a feel for active,living mathematics,even if most of it goes over your head at first.Go to seminars as often as you can.