Recommended resources (journals, blogs, etc) for nurturing a background of statistical/methodological rigour in behavioural science research?
There are various loosely-defined 'camps' among reform-minded methodologists, so you probably want to try to follow a representative of each.

Andrew Gelman is a prominent landmark in this area, and runs a really nice frequently-updated blog at statmodeling.stat.columbia.edu with frequent paper critiques.

The JASP crew also have a blog, https://www.bayesianspectacles.org/ which is a bit on the aggressive side, but JASP is great and it'll point you to things like the recent 'redefine statistical significance' paper, which will give you plenty to talk about if you run into a methods guru on their coffee break.

You might also enjoy http://daniellakens.blogspot.com/ and the author Daniel Lakens has a Coursera MOOC on this sort of thing. To be sure, there are a ton of good stats MOOCS, just this one happens to be by a psychologist with similar concerns to you :-)

This is a short list but I think it cuts across the camps I'm aware of (I'd love to know more about the camps in methodology myself, maybe some other StackExchange folks have some pointers!) My impression is that at minimum you should try to read someone who solves their problems with parameter estimation and cares about S and M errors (Gelman, Kruschke), someone who solves their problems with model selection and cares about Bayes Factors (EJ Wagenmakers 2018 "Bayes factor design analysis" maybe?), and someone who is happy to be frequentist so long as you're super careful about the interpretation (Lakens, I think? Or, like, most of statistics? Efron's awesome feats of bootstrapping are a great antidote to Wagenmakers' extreme takedowns of everything frequentist).

In my particular corner of psychology, Stan is taking over the world, if that's interesting to you you might get some good mileage out of http://elevanth.org/blog/ I reckon upwards of 20 people recommended the 'rethinking' textbook to me, and they were right! It's pretty great.