Actual Assumptions About How The Universe Works
1. There is no Cartesian Demon. Our senses may be error-prone, but in general, they reflect the real world.
This is probably the only true assumption on the list. If there is an actual malicious entity deliberately rewriting our sensory data with the intent of deceiving us, science just doesn't work. Not much we can do about that, so we just write it off as unlikely and hope we're right.
2. The world behaves according to some set of consistent rules that can at least be approximated.
This basically follows from the first assumption. We have rules that approximate the way the world works, so unless there's a Cartesian Demon screwing with us, we don't really need to assume this anymore.
Important Rules of Thumb
These aren't really essential assumptions, but they help make certain types of common mistakes much less likely, and as such are near-universal.
1. An experiment that nobody else can reproduce isn't an experiment; it's an anecdote.
People make mistakes and people lie. If the experiment can't be reproduced, we have no way to verify that one of those didn't happen.
2. It isn't enough for a theory to perfectly fit the existing data; it needs to make a prediction about something that wasn't used to create it.
It's possible to take any set of data, even completely random data, and construct a model that fits it perfectly so long as the model is complicated enough. (q.v. Overfitting). Such a model is useless since it will make incorrect predictions about any data outside the set it was constructed on. But conversely, if a model can make correct predictions about data it wasn't privy to, that's a good indicator that it will be able to do so again.
3. Sometimes weird things happen by chance
It might seem really weird to flip 4 coins and have them all come up heads, but on average, that'll happen once every 16 times you try it. As such, it's important to consider whether some weird result is statistically significant: how unlikely is it that this result would have happened by chance.
4. We will make mistakes
We're human. It's inevitable that we'll screw up. Our natural tendency when we come up with a cool idea is to try to convince people that we're right. But confirmation bias is a thing, and it's way too easy to overlook flaws if we do it that way. And if errors make it into the literature and other people try to build on them, that'll just waste everyone's time.
So first we try to see if we might be right, and then as soon as it looks like we are, it's necessary to assume we've screwed up in some horrible yet non-obvious way and do everything we can to find it ourselves. If we can't find it, then we submit it for peer-review, so other experts can look over it and find the errors that we were too stupid to notice ourselves. (And there will be some, even if they aren't show-stoppers). And if it makes it through peer review, then we assume (or at least hope) that any remaining mistakes aren't too bad, and publish.
Some mistakes will still make it through, which brings us back to the "reproducible experiment" rule. If a bogus result makes it into the literature, eventually someone will spot it, recognize the flaw in the experiment, and do a new experiment to confirm or disprove the result. And so the process continues.