I have the xbox360 and it is the absolute best around!! You can play against people and speak to people all over the world. If you get the Live Vision camera you can also see people too as well as download games from xbox live arcade. www.xboxlivepipeline.com - the xbox 1 is very outdated, If you get a 360 you will be well impressed and YES you can play xbox games on the 360 so long as you have the 360 harddrive attached.
You will however have to download a small update program and transfer it to the 360 via a Disc or Flash Memory. It only takes two minutes and then you can play all games on the compatibility list.