One good way to lift is to wear no shoes, as it adds an additional element of stability. Yes it's also not 100% safe since it would hurt if a weight drops on them, but think about the increased proprioception, muscle activation of the feet, ankles, and body, and improved coordination - the benefits outweigh the disadvantages if you are allowed to use them.
The problem is many gyms have a rule that doesn't allow taking shoes off. Then, for squatting special squat/weightlifting shoes are very useful, but I would bring a very light shoe with no drop for deadlifts (the heel lift in the weightlifting shoes makes a DL harder actually since it increases range of motion).
Don't do Olympic lifts or squats in running shoes. The only advantage I can think off is you can navigate the gym a lot faster than with weightlifting shoes, so you if you start sprinting to the rick you will easily beat the guy who just did biceps curls trying to make it there in his strap weightlifting shoes.
Another good option to build feet and ankle strength are shoes like Vibram five fingers. Anything but please don't use running shoes in gym unless you do cardio, BW or treadmill stuff!