The main thing is to minimize how much of it you bring in, in the first place. You will be amazed how much difference it makes to change your shopping habits even slightly. And now that my eyes are opened to it, even if we got trash service tomorrow, I'll be darned if I will pay companies more for their packaging than their product anyway.
We didn't last long with a burn barrel. What a mess, eventually you always end up with a rusted out barrel full of sooty crap to clean up.
Me or hubby haul our trash to work, put it in the dumpsters there. We have permission, and since it's only about 1 kitchen size bag a week, plus the trash can from his garage, it doesn't cost them any more or cause the dumpster to be emptied any more frequently.
The Elks lodge in our town has a receptacle for recycling newspaper and cardboard, we make use of it. Fold cardboard boxes flat to keep the bulk down until you take it away. Takes 2 seconds, BFD.
You can compost the biodegradable stuff. Keeps your trash from getting stinky to keep the garbage out of it, too. We have dogs who will eat about anything, and chickens, so that takes care of most of it for us. You can build a little compost bin to keep it neat looking and keep the varmints out. (As opposed to just starting a pile, that works, it just ain't pretty) The finished compost is great for garden, landscape plants, house plants.
Paper other than newspaper - run thru a shredder. More secure to shred it. Makes great nest box filler for the hen house. Have heard of people using it for mulch, but we don't get that much junk mail!
2 liter soda bottles, juice bottles, any decent size plastic bottle - rinse, fill with water, tuck in any empty space in the deep freeze. Makes it more efficient and you will always have potable water on hand. You can also use them in place of ice in a cooler. Drink them when they thaw, or bring them back and freeze them again. You can also use them for dry storage, use a funnel to fill with rice, beans, etc.
Food cans/aluminum cans - recycle. Rinse them so you don't draw bugs while you accumulate enough to make a trip to the recycler worthwhile. The food cans aren't worth much but the aluminum makes the trip worthwhile. We get about $100 every 6 months from our recycling.
Most glass jars get re-used at our house. You can't can in them, but they are handy for storage. I also make refrigerator pickles, so I use up a bunch of them every summer and give away pickles in the free jars rather than my canning jars I paid for and likely won't get back.
You can re-use butter tubs and cottage cheese containers and such, but I don't. We buy real butter, no tubs. The cottage cheese, sour cream and similar containers are just so flimsy I don't mess with them.
By now you probably think I'm weird, but I was raised by my grandmother who went thru the depression and she didn't waste a thing. She even washed and saved her styrofoam meat trays, we insulated a dog house with those and found other uses for them.