Pictured below are a few photos of pedal systems - these happen to be Shimano products but many makers offer a similar selection in the same style.
Pictured above is the most popular type of pedal system, the pedal tension is adjustable to allow for beginners to clip in easily and you can use either side , but you can only use it with a cycling shoe with the cleat fitting attached to the sole of you shoe.
Another type of pedal offers the rider the choice of either using a cycling shoe with a cleat or regular shoes. The multi purpose pedal - regular shoes to ride to work in and cycling shoes for your weekend rides etc.
The last pedal is the flat pedal, you can use this type of pedal in boots, outdoor sneakers etc and it has plenty of grip thanks to its adjustable pins.
So which type of pedal is best ?, that really depends on the type of riding you regularly do , most racers prefer the SPD type system as it looks your feet securely to the pedal and are often very lightweight. Some downhill and free ride cyclists prefer a flat pedal as it gives you the option of quickly putting your foot down when needed. Our course your pedal choice dictates what type of shoes you will need to wear. As for shoes well they come in a wide variety of prices, colors and styles. If you have wide feet you will need to hunt out some of the companies that offer a wide fitting cycle shoe , not that many options but you can find them. Also spd type shoes with cleats attached are not very comfortable off the bike as they are designed manly just for cycling not walking, but recently some makers are improving shoes so that you can walk around comfortably as well. Now if you use flat pedals like myself you have a vast array of options from sneakers, purpose outdoor shoes with super grippy soles these type of shoes are basically a hiking shoe, built tough, maybe made of goretex with hard wearing soles. Just buy a pair that you are comfortable walking around in, my current cycle footwear is a pair of North Face goretex trail running shoes . Fantastic at keeping your feet dry on a daily commute and grippy enough even in wet weather.
Pictured above is a typical pair of cycle shoes with two velcro straps and a ratchet type strap and holes in the base of the sole to attach your cleat system to lock into your shoes.
The above shoe is actually marketed towards rock climbers or hikers looking for a solid approach shoe, it is made of leather and rubber with a wide foot bed and a vibram sole. This type of shoe works well with a flat platform type pedal and offers the rider greater comfort on and off the bike.
So folks there it is the basic over view of pedal and shoe types , think about what type of cycling you like to do and go to your nearest bicycle shop and try on a variety of shoes and check out the pedal selection available . If for any reason you want to use flat pedals once again visit your local bike shop as flat pedals also come in a wide variety of prices, at present I am using a flat pedal from the BMX brand Odyssey the model of the pedal is called the twisted PC and for a cheap plastic pedal these are very well made and not expensive to replace if worn out. As for shoes for flat pedals go visit your nearest sports shop or outdoor store and find something that is comfortable for walking as it will be fine for cycling as well.