Waders & Boots!
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Chest waders are either made of neoprene material or a more lightweight breathable fabric. You can buy them with the boots attached or "stocking foot" versions which mean you'll need to purchase boots separately.
The neoprene waders will keep you warmer so if you do much fishing in colder water or colder air temperatures, you want to consider these. You can also buy lightweight waders that are big enough to fit layers of warm clothing underneath. Either method is fine.
Boots have to fit correctly. I wouldn't spend much time worrying about style and features, as they're all pretty much alike. Again, just get what fits. You will usually have to go up a full size from your regular shoe size to accomodate for the bootie, which is usually neoprene in all waders.
You can also purchase hip waders which can be easier to use for quick fishing trips. Also, they're usually pretty tough so if your in rough foilage and you don't expect to encounter water over the knees, these could be the way to do. For the small streams near where I fish the most, I usually just grab my hippers and go. They're tougher in heavy brush too.
Traditionally, the bottoms of fly fishing boots have a layer of felt which provides for better footing on slippery surfaces, like moss covered rocks. They work well, but it is now widely believed that they aid in the spread of ANS (Aquatic Nuisance Species) from water to water when you fish different locations. You can learn more about ANS here.
'Sticky' rubber soles are beginning to replace the felt soles, but they are considered a bit less effective than the felt. I'm sure there will be new developments of these soles in the near future.