Should I go into school tomorrow?

I'd stay home because all the schools panic at the smallest bit of sniffles at the moment so you don't want to be causing anyone unnecessary anxiety. There may be pupils who are scared of catching Corona and seeing you cough might make them worry even more that there's a case near them

Yeah, I'd say stay home too. It'll be easier for you to take care of yourself and probably less stressful for other people at school as well.
If you can still get things done, then there's no reason not to.

I'm in the same exact position as you: got a sore throat and a runny nose, essentially a cold (it's very common at the beginning of the school year as you're being exposed to a lot more people again: in fact I saw another user on TSR earlier today saying they had a cold too, so it's not just us two). I took the day off today, so if you're still unwell tomorrow then I'd advise you stay home too, even if it isn't Covid. It's perfectly fine to take time off if you genuinely need it, and it's probably best that you do for both your own sake and your classmates'.

Get well soon.