"Fråga" - I would use it when you ask a question to someone and that person just answers what you asked. "Ber/Att be någon om något" - This is more formal way of asking than just "Fråga" but I also believe it is used when it comes to the person you ask have to do something physical for you. For an example: "Kan jag be att få kaffe?" This is just how I would use it so it may not be 100% right! :) Lycka till!
I found the answer to my question in another discussion. The way it was explained is that "fråga" is used when asking a question, whereas "be" is used when asking a favour.
'to question' is either att ifrågasätta (in the sense, 'put into question') or förhöra (in the sense 'interrogate') in Swedish.