Bedding is important for two main reasons: enrichment and ammonia control. You need to have bedding in your rat cage. Bedding is the main defense against ammonia. Ammonia destroys the cilia (small projections lining the respiratory system, helps remove irritants) making respiratory issues more common. There are a lot of different kinds of bedding and debates over what's safe and what's the best to use.
Paper bedding (such as Carefresh or Kaytee Clean 'N Cozy) is bedding made out of recycled paper. You'll find paper bedding being pushed in places like pet stores where they primarily profit from it. You may also find it recommended in some online communities. However, paper bedding isn't very good! The biggest issue with this bedding is that it has very poor ammonia control. It's marketed as absorbent, but when it absorbs pee, it holds onto it. All that ammonia stays and builds up. This is why paper bedding goes bad very fast; it needs to be changed every 2 to 3 days, sometimes every other day. Paper bedding isn't recommended to use for rats.
The use of wood shavings with rats is pretty controversial. The three most common wood shavings you'll hear about are aspen, pine, and cedar. Do not use cedar bedding. Pine and cedar both contain phenols, which are natural oils that can be really harmful to sensitive respiratory systems. Heat treating (kiln-drying) effectively removes all phenols from pine, but not cedar. Kiln-dried pine is 100% safe to use and is one of the best beddings you can get. Aspen is also a good option and is safe regardless. Wood shavings naturally neutralize ammonia and only need to be changed out once a week. Kiln-dried pine and aspen are the best beddings to use for rats.
Some other commonly used beddings include hemp, cardboard, and fleece. Hemp is a very good option! However, it's very uncommon and expensive in the US; you'll only really find hemp in the UK or Europe. There are a few different types of cardboard that are used as bedding. Regular recycled paper or cardboard isn't suitable bedding due to the problem with ammonia control. If you're using cardboard, you need to be using corrugated or horse-grade cardboard. These are the only suitable cardboards and, while expensive and hard to find, it's a pretty good option! Fleece is a very common bedding you'll see on Instagram or Pinterest. Fleece also has a problem with absorbency. One way you can solve this is by using (rat-safe) puppy pads underneath, but even so, it'll have to be changed out every 2-3 days. Fleece also doesn't allow for digging and burrowing like loose bedding does, taking away a lot of enrichment. Fleece is only acceptable for shelves or in hospital cages (as loose bedding may stick to healing wounds and stitches). Another sometimes used bedding is soil, used in bioactive set-ups. Bioactive is possible, but it's a lot of work! If you're interested in bioactive set-ups, try joining Facebook groups or doing some research on the topic. Corncob bedding is something you're told to avoid, which is true (to an extent). It's very absorbent, so much so that it can cause mold if left for too long. If you try corncob and you notice it starts to mold, throw it out and switch it for another bedding as soon as you see it.
Never use scented bedding or bedding infused with baking soda. It aggravates sensitive respiratory systems and can cause a lot of issues. Don't use cedar for similar reasons, either. Straw isn't harmful but should be avoided because it's dusty. Pellets aren't harmful either but are hard to walk on. Pellets shouldn't be used as the main bedding.
It's true that rats can be litter trained. Many people have a litter box and fill it with litter different from what they use as their bedding. You don't need to use something different from your litter, but some people like it better that way or they say it's easier to litter train your rats when doing so. There are a few options you can use for litter, the most common being pellets and cat litter. Wood pellets and paper pellets are both fine as litter. Cat litter like Yesterday's News or any non-scented litter is usually fine too.