I’ve never focused much on footwear for working out. All I figured I ever needed were runners that had soles without holes. I have since learned that the correct shoe for the activity can mean you’re less likely to be injured. Runners for weightlifting should allow you to distribute weight uniformly so you remain steady. If you wanted to compete, you can get specific shoes for that but I just use runners since I only want to be stronger. Have a look at this post to see what I mean.
I have specific runners for running as they have an internal bump that counteracts my feet’s pronation (bowing inwards). A physiotherapist or a well trained shoe sales person (not likely to be at big box stores) will be able to identify what specific pronation issues you may have and direct you to the correct type of shoe. I highly recommend taking this route if it’s possible because an injury will sideline any work out intentions you may have.
This page can help you navigate footwear and decide what to buy for what activity.