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To start camping, it's easy. To start BACKPACKING and camping, it's a little harder since stuff needs to be lighter, and easier to carry.
To start, you need to sleep somewhere dry. I hammock camp, and love it. It's an awesome way to camp, and after years of tent camping as a kid, I'm really really happy I switched to a hammock. Pic related. It's the basics.
You can get a Grand trunk ultralight hammock for $15-$20 on amazon. I have one. It's awesome for the price. You also need suspension to hang it. Get ENO atlas straps, they're about $20, and are super easy. There are other, lighter ways to do it, these straps are just super easy.
If you want to use a tent, go for it. Get a 2 person tent even if it's just tou. There's more room in them, and a ladyfriend can join you when she gets cold. Or man friend, it's 2015, no judging. There are lots of options for tents. Even walmart has tents small enough to backpack with, although they aren't great quality, they will get you started.
You need a sleeping bag. You probably have one laying around somewhere. It's probably heavy and outdated, but hey its free. You might want to buy a mummy bag, they're lighter and pack easier. I have a 20 degree bag from Gander Mtn, called a Men's Chilkat. It was $50, goes on sale often, and is made well. A 20 degree bag is usually good to around freezing temps, they're generally cold at their rated temp. Rated temp is usually the survival temp, not comfort temp.
If you're in a hammock you need under insulation, when you lay on a sleeping bag in a hammock, it compresses and your butt can get cold easy. If you're sleeping on the ground you need something to lay on as well. A sleeping pad works for both hammock and ground. The cheapest is a $10 blue pad at waart, they go up in price from there. You get what you pay for, a cheap one in a hammock is fine since the hammock provides the support. For sleeping in the ground I like an inflatable pad, like a klimit static V pad, about $50ish.