Put your clippings into your compost, if you have one. The clippings will break down and then it can be used to make nutrient-rich soil that provides prime planting of fruits and vegetables. A compost is simply a bin in which you put biodegradable items. Clippings will break down within several days; the warmer the weather, the faster they will break down.
Bag your grass clippings and then take the to a regional composting facility. Some areas have a regional facility, but not all do. Many people prefer a mesh bag. Paper would likely break, and plastic is counterproductive because it is bad for the environment. If your region doesn't have a composting facility, use your own compost or distribute the grass throughout your yard.
Purchase a mulching lawnmower from a home improvement store. The mower, which can cost between $200 and $400 for a regular push mower, will chop the grass and then will automatically distribute the clippings over your lawn. The mulch will provide more nutrients for your lawn and promote healthy growth.