Doing the lifts with a pair of running shoes is like doing a heavy squat on a mattress. It is too unstable and you are begging for an injury. Weightlifting shoes provide a stable platform and can also sometimes provide support for individuals with tight calves. Therefore they are an essential equipment for your training.
Based on my trainees, and my own personal experience, below are a few pairs of shoes I would recommend. The price can range from $90 - $200, but these shoes should last for many years. You will only be wearing them in the gym, and not on a night out with the ladies. Make sure to go to the store and try them on to get the right size. You want them to fit just right. Once you are sure about the size you can shop online to get the best deal and save some cash.
Weightlifting belts can help increase intra-abdominal pressure when performing the lifts, this in effect allows a trainee to keep their lower back stiff to avoid flexion. It is not a necessary piece of equipment. Really strong individuals have been able perform the lifts without a belt, but if you are prone to back injuries I would suggest getting one. Think of it as insurance.
Weightlifting belts should be stiff and have no elasticity. They should have a constant width of 3 - 4 inches all the way around. Make sure you measure your waist between the top of the hip bone and the bottom rib. That is where the belt should sit.