If someone knows my bitcoin address, does this mean they can spend it or move it? What if I wrote down or saved a file with the address on and someone found it, can they take my coins?
How secure is it?
No. An address is designed to be given out so that people can send you money. You cannot spend Bitcoin by knowing an address, otherwise block explorers wouldn't be showing people's addresses. What is needed to spend Bitcoin are private keys. Those need to be kept secret and are never given out to anyone. One private key corresponds to one address.