I've been casually dating this girl for the past few months. We used to see each other once a week or every other week, however, as of late we've started hanging out 2-3 times per week. She's met my friends and vice versa and she's always very affectionate with me when in public (holding my hands, rubbing my back, etc.).

What's the best way to transition into something more serious with her? I've ready a lot of posts saying that guys shouldn't be the first ones to bring up 'relationship' talk, so should I just continue seeing her a few times a week and just assume she'll eventually bring it up?

Yes.

You're getting laid with out the title of "relationship" right now. What's wrong with that? Don't fix something that's not broken.