Wearing the appropriate footwear when training with free weights and performing compound exercises such as Squats and Deadlifts, or any type of standing overhead presses is something I always advise when I meet a new client with the goal of improving strength.
When it comes to the weight room I would always emphasize wearing a flatter sole shoe that allows you to spread your feet and give you an even balance across the sole of your foot when performing a full squat meaning at least parallel if not deeper.
The flatter-soled shoe will give you a bit more ankle mobility so you can get down and drive yourself back up to the start position. Similar concept with the deadlift–just being able to spread your feet and get closer to the ground can make a hell of difference.
The average gym user will be wearing a running shoe which is totally fine for running/jogging etc. But The running shoe has cushioning/compression system which is the support you may need when running, but not for weightlifting . Footwear I would recommend that is multifunctional and I wear myself in the gym and when doing any HIIT, metabolic conditioning, sprints, etc. and gives you lateral support, meaning when performing exercise moves where you lunge to the side or pivoting
you will reduce the risk of rolling your ankles and getting injured is http://www.newbalance.com/Minimus-20v3-Cross-Training/MX20-V3,default,pd.html
, which is available for women as well.
A cost effective option is the Converse All Stars http://www.converse.com/#/products/Sneakers/ChuckTaylor/136567F
. Other brands of footwear on the market like Vibram 5 fingers, Innov 8 can work well too, when performing lifts such as squats and deadlifts. The main thing is to purchase cross training shoe versus a running shoe as the sole of the shoe of the cross trainer tend to have a lower and even heel-to-toe drop which will work well for you if you are going to be working out doing a mixture of functional activities.
Other brands of weightlifting shoes on the market that work well Ristos, Adidas, and Do-Win, just to name a few.