Exercise (IV) is being measured as a binary categorical variable - either you do, or you don't.

Is it an experimental study? If not, then as a reviewer I would not accept just 2 broad (and perhaps ill-defined) categories. Why not measure excercise (sports, using the stairs instead of elevator etc.) in more detail? Instruments exist which are designed for measuring such sort of thing (at least I have encountered some, as study subject).

Quality of Sleep (DV) - was assessed by asking the question: "How many days did you have trouble falling asleep, staying asleep, or sleeping too much over the past two weeks?"

Well-designed instuments exist for the assessment of sleep duration and sleep quality. Why not use one of these?

Respondents answered "none" or a number of days between 1-14. I am diving the answers into three categories: "none" signifies a good sleeper, "1-7 signifies a moderate sleeper" and "8-14 signifies a poor sleeper."

There is no reason for categorizing. And you have given no justification for your arbitrary categorization. Does empirical evidence exist for this? If not, why not just stick to "number of days". Categorizing detroys statistical information and can lead to artifical or wrong results.

Age (moderator) is being stratified into three categories: ages 18-39, 40-64, and 65+

Same problem as with "number of days". Someone who is 64 years old is considered similar to a 40-years-old person, but dissimimilar to a 65-years-old. That make not very much sense. And it produces difficulties for the data analysis (see below).

I am attempting to figure out which type of statistical analysis tool to use - ANOVA, logistics regression?? We have not gone over these in my research class yet - I have attempted to do my own research, but I'm still unsure - please help!

I suppose this is not a real research study, more of a homework problem? Your variables up to now are poorly defined, I'm afraid, poorly measured, and poorly treated afterwards through stratification. Besides, if you treat "sleep quality" as an ordinal variable with 3 levels, it will be a bit difficult to perform a moderation analysis (you would then have to make yourself familiar e.g. with ordinal logistic regression).

So if you use "number of days" as dependent variable, you could use multiple linear regression for your analysis. Preferably with original age in years, and with a better measurement for "amount of excercising" (but if you stick to excersise yes/no, then you can use this as a 0/1 variable in regression). The moderator effect is represented by the interaction between age and excercise (if you had 3 age categories, this would be more difficult to perform).

Just my 2pence

Karabiner