Here's the most important thing for you to know.
Most shoes are not going to do good things to your feet.
Generally, dress shoes, sandals, and classic shoes are bad. Classic shoes are what most people think of when someone says "sneaker". It is a shoe made by a company that makes athletic shoes, but has stripped all its technology out of a shoe, put in basic cushioning, and called it a classic. Dress shoes, sandals, and classics give your foot a very small amount of cushion, and nothing else. High-heeled sandals and dress shoes are even worse.
There are some sandals and dress shoes that use custom cork, running shoe cushioning, or molding insoles to provide better cushioning. They are still not as good as having the right running shoe.
A running shoe looks like a sneaker, or even a classic. Running shoes are categorized as follows.
Neutral: Made for someone who does not have overpronation or supination (underpronation). It gives relatively good cushioning and shock absorbtion, through gel, thick EVA, SPEVA, ABSORB, waveplates, "pearls" or some other technology. This is a good, cheaper, lighter, running shoe.
Cushioning: This is almost the same thing as a neutral shoe. The words can be used interchangeable. A cushioning shoe, however, typically has a lot of cushioning. It uses the same technologies, and is a good shoe for someone who has a flat arch, supinates, or has a high arch (but does not overpronate).
Stability: A shoe that has a medial post (a portion of the inside sole that has thicker, denser material), or some other stability technology. Stability helps overpronaters arches from flattening too much. This helps them not turn their ankles or knees too much, and guides the foot into a good toe-off.
Motion Control: A stability shoe souped up. Typically heavier than a stability shoe, with a massive medial post, the motion control shoe is for a serious overpronater. It gives lots of stability, which could also be called motion control.