What is wrong, my new Xbox or my internet?

that sounds like your connection. i share a connection at my house and when i play black ops and no one else is online there is no lagging whatsoever, no if someone is online in the living room and on a laptop and on the other xbox...its very very laggy also my friends get kicked out a lot if multiple people in their houses are using the same connection as them... my suggestion would be to get your own internet provider and make sure you have a security key on it so no one else uses it... hope this helps