If the disc game is on the list, put it in an XB1 and it will download the game to the hard drive. 360 games don’t play from the disc on the XB1, but the disc needs to be in the drive (it’s the same with XB1 games).
If you have a digital license for a 360 game, it will show up in the “ready to download” section of your games & apps menu. You have to be using the gamertag that the game is associated with, here’s hoping you don’t have digital licenses spread across multiple tags. If the owning account designates that XB1 as its “home” xbox, then all users can access the games.
360 saves have to be moved to cloud storage to work on the XB1, so keep the 360 around until you are done copying. All saves are backed up to cloud storage on the XB1, but you need XBL Gold to access cloud storage on the 360. Once you do, simply copy any save from the 360 to cloud storage, it will show up as a storage device in the 360’s menus. There is no way to connect a 360’s physical storage device (HD, formatted USB, memory card) directly to an XB1 that I’m aware of.