Yup. The neuroimaging folks are really good about sharing data. If you want to look at MRI/MEG/EEG data,
https://openneuro.org might be a good place to start.
CRCNS (https://crcns.org/data-sets) has some neurophysiology data (i.e., from implanted or inserted electrodes). This sort of data is shared a little less often, in part because it's often acquired and stored in weird, homebrew formats, though that's slowly changing.
ModelDB (https://senselab.med.yale.edu/ModelDB/) has a large collection of computational models. These are mostly biophysical models, though there's some other stuff in there too.
Depending on what you're looking for, there are other more specialized repositories. NDCT (https://data-archive.nimh.nih.gov/ndct) has mental-health related clinical trial data, though you'll have to do some paperwork if you want subject-level data, which is fairly common for clinical data. MIT has a collection of eye movement data sets: http://saliency.mit.edu/datasets.html
As for the best methods, this comment box is far too small to contain all my thoughts on that :-) It depends on your question and experiment. Sometimes, all you really need is a t-test (or the randomized version), but that requires getting the experiment just right. Other times, you might need a morass of signal processing and dimensionality reduction, fed into some giant multi-level Bayesian model in a vain attempt controls for all the stuff you neglected when you designed the damn experiment. Happy to send you some pointers if you have something specific in mind though!
Physics has historically had a huge leg up on the other sciences because they had real models that made testable, quantitative predictions. We're finally starting to learn enough about the brain that we can do this for neural data too, and I'm really excited about that!