I think I see two different questions. The first is how do you save money on feed with free range birds?
That one is hard to answer, partly because I don't know what your forage looks like or what your experience or expectations are. Also I don't know where you are so I don't know weather or other considerations. But I can maybe talk around that.
For thousand of years people like small farmer kept free range flocks that fed themselves in good weather. For that to work they need a pretty good variety and quantity of forage. A manicured back yard does not cut it. They need a variety of grasses and weeds plus places they can scratch for and catch creepy crawlies. It helps to have big animals like sheep, horses, or cattle they can follow behind. They can get a surprising amount of nutrients scratching through poop. They supplement food during winter, typically with things they grow instead of buy.
These people were not raising chickens for show. They did not raise the decorative chickens. They did not worry that much about breed, typically they were barnyard mixes that probably had a lot of game in them. The chickens were not huge and did not lay double extra huge eggs. They were designed to forage for themselves. Predators were a huge issue. People will tell you that you can't do this. Most of us can't because we don't have the right conditions. But those of us that grew up on farms like this or are still living like this understand it is possible.
I don't know what conditions you have with your flock make-up or your forage conditions. Typically you reduce costs by giving them as much quality forage as you can, giving them kitchen and garden wastes, and growing what foods for them you can. The farm like this that I grew up on we grew corn for the milk cow, plow horses, and chickens.
Now the question on bedding. You don't have to buy bedding. People are using dried grass clippings, dried leaves, hay, straw, wood chips, basically what they have free or cheap. Dad used a dirt floor in his coop.
There are issues with the coop. A wet coop is a problem. If you can keep your coop dry life is good and easier. If it is wet you have to work hard. So keep it dry.
Another issue is poop build-up. Part of this is chicken density. How may chickens do you have pooping in how big an area? If you have weather or other conditions so your chickens can spend all day every day outside they are not pooping in the coop. Depending on your weather free range can really help with that. If they spend a lot of time in the coop poop can build up. Same thing can happen if you have a small run.
They are inside at night and they poop a lot at night. They are not moving around so the poop can really build up under the roosts. You probably need to manage that poop. As always there are different ways to go about that. Some people use a version of the Deep Litter Method and turn their coop (or run) into a compost pile. That poop gets mixed in with the bedding and composts. Some people clean out under the roosts when it gets bad. Many of us use some version of a droppings board, put something under the roosts to make it easy to get the poop out from under there.
Some people manage this by changing bedding on a regular basis, maybe weekly, especially people living in suburbia with a small number of birds in a small space. You do the best you can with what you have to work with. As you can see from the above responses some go a lot longer.
I use droppings boards to remove poop from under my roosts and put that on my compost pile. I keep my coop so dry it doesn't even compost. I use wood shavings, topping them off when the chickens have pretty much scratched them to powder. Mine are never in the coop except to lay eggs or sleep. About once every three or four years I empty my coop floor into the garden in the fall, by spring it has broken down enough to not be a problem at planting time. I'm at an extreme, I probably should do this every fall for the good of my garden but I can be kind of lazy sometimes.
The bottom line is how often you need to clean out your coop floor depends a lot on your management but also other considerations. If winter keeps your chickens in the coop a lot you probably have to work at it harder. We all have different conditions.
