Opportunity to change the way we test

I have an opportunity to dramatically improve and standardize how my small company tests new code. There are multiple teams that each tend to specialize in their product. We're ostensibly an "agile" shop but it feels more like "freight train" SDLC where "it's going out dark" is an excuse for delivering bad or insufficiently robust code.

Can anyone offer any advice for me in this role? Best practices, mindful improvement of processes, STANDARDIZING processes, etc.