Let us dig right into it with the worst choice, running shoes.
You may ask what’s wrong about it?, and here is the explanation.
Running shoe tend to have too much cushion and often bring your toes up.
The cushion will compress as you lift and it may very often lead to instability.
Worst case scenario? Injury due to the instability.
Most times it will just mess up your technique/form and lessen the strength you can apply to the motion.
Obviously this is not what you want, so what should you do about it?
Here are the options we would recommend:
Cheapest option ( good one if your skeptical too ) :
You like how cool your running shoes look like, don't want to carry fifteen pairsto the gym, and may be doing some cardio on the treadmill?
Just get barefoot when doing squat, lunge, dead lift, or any motion that requires stable support,
you might be surprised by the difference in stability and strength you get.
Not a fan of the workout barefoot mode? ( or the gym staff comes running to you the moment you take your shoes off T_T )
Most convenient option:
A regular pair of shoes like Converse or Stan smith, something quite flat without too much cushion.
Obviously won’t be great for running, but it is a pair you will be able to use outside the gym too.
Most versatile option:
Crossfit / training shoes would be the most versatile option, and should still allow you to run with newer models.
There are plenty of choice nowadays, such as Nike Metcons, Reebok Nano, Under Armour, etc, 
Lastly, if heavy lift is your main focus, dedicated weightlifting shoes is a great option.
They lift the heels higher than usual and provide great stability and position, especially for squats.
A great work around if your gym space has it, is to use a fitness wedge to reproduce the stance those lifting shoes provide.
That's it for us, and wishing everyone still here a great day!