I understand this is a broad question but I'm taking the risk to ask anyway. Robotics, from what I can tell so far, is a detailed, diverse, and thorough field. However, if there are better areas of research to invest time into, what would those areas be?
Yes this is a broad question, you answered it partially is depends on your project.
If you are working on a team, you probably would be responsible for the electrical, or mechanical, or navigation, and goes on, so you can go more deep inside the topic.
But, anyway, you should know sufficient about the other team topics to integrate things together.
If you are managing a project, working not in a big team, or want to get more specialized, there's so much things related to robotics. And as a base I would say: Electronics, mechanics, electro-mechanic, programming. This will include for example in mechanics, all the basic concepts, tons of literature and the experience you gain with time, and this includes how the things you study are manufactured. That includes (but not only) how the material is cut, welded, molded, each type of the processes and their pros and cons, etc.
Kits will help with this, but it will take some years to get all together, as the kits have the pieces manufactured, you probably don't experience the more manufacturing questions. You would not get complicated by "will this method of joining" two materials will be the correct choice for this type of application? Will the stress on it be a problem?
This is just about mechanical, now take that to electrical, electro-mechanical, and programming, as you see, it takes time and effort. This means you would not know all about just one, you probably want to specialize in one subject once you get a project that will require it, but with the base and experience in this topics, it should be a more easily learning curve to get on the subject.
As robotic is a wide topic, you probably won't get all in small time, this would go with the experience, literature, already implemented solutions, and goes on.
Arduino really helped me get started. Tons of support and great products these days.
Parallax is another good company offering great starter kits. PBasic is a good starter programing language which runs on the Basic Stamp. The Activity Robot Kit was very useful for me.