The big three is a term used in the backpacking community to refer to your pack, shelter and sleep system. These three will likely take up the bulk of your gear budget. There is a delicate balance between cost, comfort and weight/pack size. And finding that balance will allow you to spend many amazing nights out in the wild.
Your shelter is one of the most important pieces of gear you will carry. It is your home for the night and will protect you from insects and weather. This can range from a $10 tarp, to a $1,000+ mountaineering tent. Some people prefer to carry a hammock for their shelter, just make sure you have a plan for rain. Your shelter choice will also inform your decisions for your sleep system.
A well planned sleep system will keep you warm and comfortable in any weather. And the quality of your sleep can make or break the next day’s hike. Your sleep system consists of a sleeping bag/quilt, sleeping pad and a pillow of some kind. Most people bring a down or synthetic sleeping bag. Down compresses more easily but is more expensive. However, if down gets wet it loses its insulating ability. An inflatable sleeping pad will save you space but a closed cell foam pad will save you money. As for a pillow, you can bring a dedicated inflatable or compressible pillow or throw your extra clothes in a sac and use that.
Your backpack is where you store everything while you are moving from one place to another. Everything you bring MUST fit in or on your pack. Your pack could be anything from a stick with a bandana tied on the end of it to a highly sophisticated internal-frame pack that is custom fitted to your body.
There is a lot to consider when purchasing your big three. It may be advantageous to borrow these if you are able to until you have decided that backpacking is for you and have a good idea what your needs are. There are also stores that sell lightly used gear that you can buy while you sort out what works best for you.