I learned a bit about ANOVA as part of second-year research methods course in undergrad psych. The first paper I took that was directly focused on stats was called Multivariate Data Analysis. It had a brief mention of ANOVA in the intro, then multiple linear regression as one of the four major topics (along with factor analysis, MANOVA, and discriminant analysis). So my experience is along the lines of ANOVA > then regression.

I don't necessarily think that's best though. It would make more sense to me to learn the big general framework first (i.e., linear models). Then later learn things like some of the specific complexities that come up more often in the ANOVA context (like planned comparisons and repeated measures).