Cages

Your hedgehog's cage should be completely solid-sided. Hedgehogs are very clumsy and poor at climbing, but they don't seem to know that and often try to climb anyway. Generally a climbing attempt will result in an immediate fall which could cause broken bones. Narrow-spaced bars (like in Hamster cages) are dangerous because hedgehogs tend to get their legs stuck in them. Wide-spaced bars (like in rabbit or guinea pig cages) are even worse because hedgehogs can easily get their heads stuck in them, potentially causing a broken neck or even death. While some rabbit and guinea pig cages have a few inches of plastic before the wire, they still aren't fully safe as hedgies will still be able to make it up to the bars by propping themselves up on their igloo or wheel. Make sure that the bottom of your cage is solid too as hedgies will hurt their feet walking on the wire floors that are in rabbit hutches. You also want to avoid any cages with levels because due to their poor eyesight and lack of depth perception, hedgehogs will walk right off any ledge. Levels are only safe if both the upper level and ramp have borders around them. The best cages are plastic totes, glass aquariums, and C&C cages (with high coroplast sides that go up at least 10 inches). 

Wheels

Bedding

Exercise wheels are a necessity for pet hedgehogs. Just like cages, they should be completely solid. Avoid all wheels with wire bars or mesh because legs/nails can get caught in them. You also want to avoid sandpaper wheels as they will cause a hedgehog's feet to bleed.
The best wheel by far is the Carolina Storm Wheel. They are completely silent, easy to clean, can be purchased with a litter pan to fit under them, and are handmade to be the perfect size for hedgies. You can only purchase them online but they are well worth the wait and cost because unlike most pet store wheels, you will not be kept up all night while they run and will not have to spend forever scrubbing poo off of ridges. Kaytee Comfort Wheels (12 inch for adults, 8.5 inch can work for babies in the first few weeks though) are usually the cheapest option and are available at most pet stores, but they tend to be noisier and harder to clean. The flying saucer discs (we recommend the 12 inch size) are another safe option, but can be easily broken if your hedgehog is on the heavier side. While some hedgehog owners use the plastic Silent Spinner wheel, we don't recommend them because there are open spaces between the two pieces that a hedgehog can get nails caught in.

There's tons of options when it comes to hedgehog bedding. Wood shavings like Pine and Aspen are low-cost and work fine as long as the bag says "kiln-dried". Carefesh paper pulp bedding, paper crinkles, and Yesterday's News pellets are all safe options as well. You can use newspaper or shredded documents as long as there are no staples. If you are successful in litter-training your hedgie, you can just have fleece liners and puppy pads instead of bedding. Fleece is the best fabric to use for this purpose because it does not have frayed edges.
Avoid cedar bedding (it's toxic to all small animals and can cause death), cat litter (clumping or non-clumping, either way it's too dusty for a hedgie to inhale), and towels or other fabrics that have loops and frayed edges that nails can get caught in.