I was given an account number and a route number from one of my friends and she said that I can walk into any bank and withdraw funds that she owes me. I don't have any bank account at all. So my question is can I go into a bank with identification and hand them the account number and route number to withdraw my funds even if it's not her bank?
No, you cannot do that. You can't just debit someone because "he said so".
You'll need a document in writing, that would show the routing number, account number, the amount to withdraw and who is allowed to withdraw it. This document must be signed by the account owner and dated.
Usually, these documents are called "checks". But legally speaking, in most countries you can write this on any piece of paper and it would be valid.
But better use a standard check.
Once you have a check - you can cash/deposit it in any bank, that part is true.