It is hard to get a good overview of all of mathematics. The best really is to take a good broad variety of classes. This you usually do (to some extent) the first couple of years in graduate school. Here you will learn the basic language of the main areas of mathematics. Here you will be get to meet various professors.
For now, then, I wouldn't worry too much about what field you want to go into. If you don't already have an interest in an area, then it is hard to say much that can push you in a certain direction. You might be thinking that you want to find an area so that you can go to a school that specializes in that area. Again, don't worry too much about it.
One thing that you probably can decide on now is whether you want to do applied mathematics or pure mathematics. Some departments are very strong in one and not the other. It sounds like you are more into pure mathematics.
If you are wanting to do a Ph.D., then I would also point out the importance of having a good advisor. Some are highly motivated and work hard on their own, but others need good guidance. If you take an area like algebra, then you have to remember that this area is huge. There are many subareas of algebra and what you will end up doing research in will (usually) depend heavily on what your advisor does. I say this to point out that the choice of advisor for many is almost as important as picking a good area. (Again, others already know what they want to do and just need an environment where they can work in peace.)
About conferences. Ask your university/department about funding to go to MathFest or the Joint Meetings. Do you have a Mathclub at your school that might be able to help you? Next week happens to be MathFest and here many undergraduate students will present work they have done over the summer (or past semesters). These talks are a great introduction to various areas. The level is low enough that an advanced undergraduate student should be able to learn something. There are also student activities at the Join Meetings in January. Talk to your department about this now!
Another great suggestion is to attend an REU (Reseach Experience for Undergraduates). These are usually summer programs where you get to go into depth on a specific topic. Usually there isn't much of a requirement for background). Google this and start to talk to your department about how to apply.
Lastly, try to talk to as many people as possible about what they do. It sounds like you have already been doing this, but ask each professor in your department what they work on. Talk to graduate students about what they do.