Training without shoes I know with squats and deadlifts a lot of the lifters that have been lifting for a while wear chucks or powerlifting shoes, pretty much flat soled shoes which are not like the everyday sneakers. I just saw some new balance minimus shoes that are pretty flat that would be good.
Training without shoes I used to, until i got the funk in them (I would run across highway/parameter ditches) Its been around a couple months since i've last seen them but i will probably bust them out again this summer just for lifting.
Training without shoes I lift barefoot more often than I lift in shoes. I like the stability and increased proprioception of foot to floor, personally.
