I just bought a new mac and set it up recently. I had all the sharing options on to help me transfer files from my old computer. I turned the computer off and went on a week long trip, but accidentally left the old computer still on and connected to the internet.

When I came home the new computer was on and no longer required a password at login. I found this suspicious and strange.

I did not turn off password required at login.
Can someone have taken over my computer while I was gone?
I have an ex who is capable of this, but I am not sure if he would bother.

Another computer on the internet?

What should I do or be looking for?

Or should I not worry?

EDIT: PS. I have turned off remote sharing, changed the password, disabled the guest account, and turned on the firewall, including blocking pings, etc. But too late, perhaps.

EDIT2: I have some logs on the machine and I don't see any sign of activity on the days I was gone. This is reassuring, but how can the requirement for a password at login been turned off?

I don't think your login settings weer changed remotely. Who has access to your home could you have left the computer logged in while others were around? We all think that it was some evil for but the real story is it is usually a computer left logged in while the operator is out of site. The firewalls and other step[s you have taken are good, remember information security begins with the user and then physical security. Good luck I hope you enjoy the Mac.