Psych's a hugely varied field, as you alluded to. You can do studies with fMRI (not really applicable) and you can do studies with a pen and paper. The real trick is what you want to ask, and knowing how to ask it correctly. Some of the most interesting studies, however, don't necessarily use either. The downside (and probably why many don't consider psych a science) is that although some methodologies appear simplistic, yet the statistical analysis that going into are incredibly demanding.

It's possible to create studies that use stuff you learned at high school (means, standard deviations, t-tests) and learn some other weak statistical tests that are incredibly useful (chi-squared, for instance) but beyond that you're going to need some genuine schooling in other methods including ANOVA and Regression.

Personally, the manner in which I like to employ psych as a citizen scientist is to find a problem, look at the research, then see if I can apply it. For instance, increasing compliance in others is a perfect example. The downside is that, as a citizen scientist, you don't really have a control group - and so it can be difficult to tell if you actually are getting an effect. Though this might not be the format you're looking for.

In terms of pen and paper stuff (sorry if I've been off topic) there are some very interesting ways to approach asking the right question and getting a valid answer. You need to know what you want to ask, in very clear terms; and you have to know why you're asking it. If you have a hypothetical question, I'd be happy to address as best I can.

In other instances methodology is key. Let's say you want to train your pet, a few simple pavlovian principles can be applied endlessly to many causes. Training your pet is the same as training your workmates, or partner. That itself is endless fun (and you're not necessarily constrained by ethics committees, either!).