I got the Pfizer vaccine and my immune system is completely fucked as it is because I'm chronically ill, so take this with a grain of salt, BUT:
After the first poke, I "just" felt like I'd been punched in the arm for a few days. I was actually surprised by how well my body handled it, all things considered.
After the second one, I initially didn't feel any worse than I already do as a baseline, but then the effects of the vaccine and the post-exertional malaise (PEM) both hit me like a truck the next day. It felt like I had the worst flu ever, x100. I don't want to freak you out, as I'm about 100% sure that my severe reaction to it was largely due to my illness/immune system, because that's just how it is (and I would have felt absolutely horrific afterwards anyway, as this kind of physical reaction also happens even when I stand or talk for too long, so it was difficult to tell what was caused by the vaccine and what was just my body doing what it does). I'm sure if I were even somewhat healthy it wouldn't have been even a fraction of as bad as it was, but for me personally, I wanted to die even more than I already do, which I didn't even think was possible.
With that said, in general, reactions to the vaccine tend to be highly variable, based on what I've read and heard from others. I know a few people (all decently healthy as a baseline) who got AstraZeneca, and their responses to the first vaccine were all just an extremely sore arm, like you mentioned, but for the second vaccine it was all over the place – one person was completely fine, one felt like they had a touch of the flu, and another one felt like they had the flu AND was puking their guts out... but within a few days they were back to normal. In general (and again, just from what I've heard and read; I hope I'm not talking bullshit), any side effects that occur seem to come on within 18 hours or so, and they tend to be stronger after the second round (which the doctor/nurse will usually tell you beforehand), but they seem to go away within a couple of days on average.