When I am registering an account at an online casino, I am asked to give my phone number, address, date of birth and my real name, should I put real info?

If you want to be able to create an account, yes! They have to be able to confirm that you are of legal age and a real person. If you use phony info they will close the account.

If you have any concern about legitimacy of the site then don't open an account.

Yes, since if the site is actually real and not a scam, if you actually win, you will need that information to collect.