I can't say I know enough about testing.
Do we care about posted times and speeds?
I'm sure they are running numerous testing programs. Just wondering what to really read into during testing.
Anyone have something specific they keep an eye on?
I am no expert, but you can't pay attention completely to lap times, as we know know the fuel loads. There will be some places where you can find laps that are stitched together when they do long running that might give an indication who is fast. First day is ususally all about getting the kinks out, putting in laps. Obviously each team has their own run plan, that dictates what they do. Typically, there is one session where everyone will put on the softest tire and goes all out for a lap, but you can't read into it to much as again you don't know fuel loads.
I expect this year to start like last year ended, there aren't that many changes. RB will be fast for at least the first few races, Ferrari will be just behind, and Merc is a wild card, but with probably the most changed car, they will be behind. They did get a good amount of laps in. Merc have also historically played this stuff close to their vest in the past. They won't show their pace until Q3 of the first race. But that is pretty true for all team, except Ferrari. historically they want to put down a fast time so they can point to in their very Italian way. But maybe with new leadership this year, they are being smarter with there testing.
TLDR - its fun, there isn't much a casual fan can glean from it. Trust nothing.