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Find is usually used when you find something like and object or something like that.
For example:
"I've found my lost wallet."
but is:
"Scientists are trying to find a cure for cancer"
correct?
If so when should I use find and when should I use find out, not just in this context, are there any rules about when I should use one or the other?
Yes this is correct. Don't use "find out" in this example.
Scientists are trying to find a cure for cancer
"a cure" is a noun, and it's a thing which can be found. It may not be a physical object, so to speak, but it's still a noun. You can nearly always use to find + noun. One can "find love", or "find happiness" too!
"To find out" means "to ascertain/discern" or "to discover information (about someone/something". There is a sense here of attempting to get to the truth, or find a definitive answer/information about someone/something.
"Find out" is usually followed by the words: that, if, how, when, where, what, who, why, etc. Sometimes "that" can be omitted, but it's still implied. Also note that "find out" is quite informal, but it is commonly used in everyday English.
For example:
It took me several days to find out [that] she was sick.
They want to find out who he is.
The government must find out why this has happened.
The police are trying to find out if a murder has been committed.
Astronomers are trying to find out how old the universe is.
It's also possible to say "to find someone/something out". This means to reveal the truth about someone/something.
