Is it a good idea to join a second group as an undergraduate researcher if I don't feel challenged enough?
Is attempting to join another group a bad idea? Should I instead tell the group head how I feel, and see where to go from there, or would that be far too disrespectful?


These two options are not mutually exclusive. You can do either, both, or none of them.

As you see it, you believe your choices are


Talk to your current supervisor - You should absolutely do this, regardless of what you decide to do with respect to the other group. Your mentor has no way of knowing what you're thinking unless you tell him; it's up to you to let him know if you need more from the supervisory relationship. Of course, you should do this in a respectful and adult way.
Join a second research group - You should consider doing this if you're interested in the research the second group is doing. As an undergraduate, it's to your benefit to experience different kinds of research, different advising styles, etc, to help you understand what you need when applying to graduate school. Note, however, that research is very time- and energy-consuming, and if you spread yourself too thin, you will experience "research" only on a superficial level and won't get much out of it. Only you can determine whether you really have enough free time and energy to participate in two research groups in a meaningful way.


Finally, I'd like to comment on your statement


  ... all of this was essentially on my own. As a result, I feel that any average joe with minimal work-ethic and intuition could thrive in a group like this.


You seem to think that because your mentor gives you a lot of independence and doesn't tell you exactly what to do, that standards and expectations in the group are pretty low. That's not necessarily the case. 

The ability to work independently is a highly valued skill in supervisees, and it's great that your advisor has given you the chance to show your capabilities in that respect. This means that he'll be able to write strong recommendation letters for your graduate applications - much, much stronger than a supervisor who could only speak to your ability to follow focused, specific instructions.

Research is not like coursework. It's supposed to be highly self-directed. If somebody is telling you exactly what to do and what to learn next at every step, you're not doing research. If you feel like you're not sufficiently challenged by what you're doing now, you are supposed to take the initiative to speed things up.

In other words, don't confuse a hands-off mentoring style with low standards. If the group in general is productive, does good science, and writes solid papers, then standards are what they're supposed to be. From your description, you are learning a lot of new things, starting a new experiment, finding out how research works, and overall describing a pretty excellent research experience.