Perhaps it's time I get a proper subscription for the TV coverage and watch it like you say over the weekend.
Try watch a few more races and even check out the F1 subreddit /r/formula1 - its a good community that share plenty of news, clips from new and old races etc - and if you have any questions, they're all very helpful
I typically just watch qualifying and the race. But it's a time thing. Qualifying if I fast forward through things still takes about 40 minutes. The race is typically a solid 2 hours or so. That's nearly 3 hours each weekend it's on and we haven't even dove into the pre-race and post-race bits or the practice sessions.
Things get interesting, like in yesterday's race, when penalties start playing into things (which resulted in different podium finishes yesterday) and some of the technical issues that happen (of which there were several during the Austrian GP, as only 11 finished the race).
I'd say yesterday's race had a fair bit happening in it, but not super duper exciting stuff.
The big part of the problem is, at least so far this year, Ferrari is once again behind the eight ball, Mercedes is (once again) running the pack, and Red Bull are sort of somewhere in the mix there.
The more exciting parts of the race are the team placing 4th, 5th and 6th. Those are typically much more exciting, but they rarely air much of those parts of the race.
r/formula1 is a good place to keep up to speed as well.
Also, I've found a few podcasts that I listen to during the week that help me stay up to date with the storylines of the season. In particular, Missed Apex[1] and Shift+F1[2] are my go-tos. The former is a little bit snarkier and opinion-based, the latter is more straightforward and factual - both are good. Check out their race previews/reviews, it will give you some things to pay attention to when watching a current race.
[1]: https://missedapexpodcast.com/
[2]: https://www.f1.cool/
A few years back, the free-to-air channel, Channel 4, lost the exclusive rights to broadcast the live events. Sky bought them up. Nowadays the Channel 4 offering has become a highlights package, and frankly that's been a pretty good outcome for me. Actually, I don't mind the reduced coverage of the actual car-on-track phase, but it's also not as in-depth on the pre-/post-race analysis, which was where a large chunk of the interest, intrigue and drama occurs.
Also, being able to record using a DVR means I can just tune in whenever it suits me, even if I'm dipping in for 15-30mins over a number of days. At least there's often two weeks between races.