Just moved into home that was vacant for some time. When I turn on cold water or flush toilet, I get a loud vibrating/humming noise in my basement - not near boiler or anything. If I turn water back on briefly, it stops. Help?

Noise from vibration's mechanical. Something's moving against something else. It could be a washer that's broken vibrating against the seat as the water rushes by or a small pebble that got caught in a valve.
Try this; Start the noise and go down in the basement. Follow the noise to where it's the loudest. Now take your hands and see where the vibration's the strongest. Let me know what you find. Good luck, tom