Lifting weights with no shoes is fine. Running shoes is a bad idea.
In a running shoe The air and gel is good for reducing the impact shock from running. But not good for lifting. Running shoes limit your strength & prevent proper lifting technique, they are unstable, the Soles are squishy so this makes it harder to use proper technique and lift properly, they are dangerous especially with heavier weights, since the soles make it hard to balance and you will be less stable.
At this stage dont get worried too much about getting the best lifting shoes. I wear nike Sbs which are flat and have no compression so they are good for being stable and lifting weights.