I feel that I have a weak application. Should I take a gap year before applying to grad school? A little about me:
-I'm a math major.
-I'm going to be starting my senior year this fall.
-I have the credits to graduate early, at the end of the fall semester.
-I have taken a few graduate level math classes, computer science and statistics  classes, and science classes (psychology, biology, chem, physics).
-I'm leaning towards applied math.
-I go to my state school, which isn't very selective or prestigious.
-I have a 3.9 GPA.     

I haven't had any advising beyond getting me to fulfill my graduation requirements and now it's my senior year and I still don't know what I want to do career-wise.  I think I want to go to graduate school because I really enjoy the school environment and learning, but I'm not 100% certain. 

Assuming I do choose the grad school path, I feel that my application would be weak at this point. I would like to stand a chance at getting into a more prestigious program than my state school. 

-I have not taken the GRE General or GRE Math test. These seem to be the most important factors in admission. I don't know that it's possible to get a good score on both between now and the application deadlines around December.
-I have no research experience yet. I plan to assist a professor with research in the fall, but I wish I would have more prior to applying.
-I have little work experience so my resume isn't impressive.
-I still don't know what program/research area I'd want to go into. I think this would show in my personal statement and recommendation letters.    

Do I have time to improve all this before the application deadline? Should I take a gap year to see if I can get some research/work experience, make sure I want to go/figure out what I want to research, and do well on the GRE? Does a gap year weaken an application?
I think you are overemphasizing the importance of the GRE. Consider it as a box you need to check, rather than "the most important factors in admission". Still, you should get started on that process only because opportunities to take the test are few and far between, and if you are ill the day you intend to take the test or otherwise do poorly, you might want a second chance.

I think you still have time to get some research experience. I don't know what is available in your field of study and at your institution, but in my field I would recommend you find a lab that would possibly be interested in hiring you on in the spring after you graduate, assuming things go well in the fall. Although applications will be due before that spring semester, at least you will be demonstrating your interest in research, and hopefully you can get a decent letter of recommendation from your research advisor.

I would also suggest finding a faculty mentor: not an administrative academic advisor, yes, those people are mostly interested in your undergraduate progress, you need to talk to professors who have gone through the grad school experience, who are involved in admissions for graduate school, etc (of course one of the disadvantages of a smaller state school is that no one there may have experience with graduate admissions, except as a student).

I don't think there is an either/or question here. If you are set on doing graduate school, you should apply and plan for what you might do to make your application better next year if you don't get into a program you like this time.