It is better not to pre-focus on concrete metrics too early. Better have the mindset of a scientist trying to figure out an unknown phenomenon. Observe, look for patterns, ask the what-if questions, form hypotheses, try to verify them, etc. Not my direct experience, I just once helped a guy who started with this approach and needed some better scripts to gather and analyze more data. This is fun stuff, everybody should try it.

It's getting a feel at first for how people visit the site, what obscure search terms are used, where they go. Do the "visits" come from real browsers (are images and JS being loaded)? What bots are coming and can they navigate the site too?

Obviously this is not measurement by itself. At that company I wrote a suite of very detailed measurement tools, and we used commercial JS-based tools too (Nedstat, Urchin). My point though is you need to understand the logs first to even know what the tools are telling you.