Tomorrow I'm supposed to sign a contract for my first "real" job ever, so I'm not sure how this works. I want to fully read the contract, but it might take me some time (like half an hour). Is that accepted or will that seem weird to them?
They should not be surprised at all. I would expect that the company will send you a draft version few days prior so you would have time to think about it and seek legal advice. If you didn’t have such opportunity and you see some points you are unsure off you shouldn’t be afraid to ask for few days to get lawyer to look at it.
Welcome new user.
Really, more normally they would give you the contract not on the same day. They would send you the contract (perhaps, the previous day - or at least that morning), telling you something like "let's meet to sign it at blah time."
To be honest I'm not sure how I'd deal with it, if you're in a more junior/starter position, and they're sort of "handing it to you to sign there and then."
Additionally, as a note, these days it's rare to sign on paper such a thing - they just PDF you usually.