Whoa, I have to disagree with a couple of the above posters.
1. "Don't anchor it to the concrete" Uh, no. The posts need to be anchored in so the thing doesn't shift. There are Simpson post anchors that keep the wood from coming into contact with the concrete, to avoid rotting the posts
2. "Use deck blocks." Not unless you live in a year-round mild climate like California of Florida. In places where it gets cold, you need proper concrete footers.
To extend onto the grass, you need to dig holes and pour the above-mentioned concrete footings (sonotubes work great as forms). Depth depends on your frost line, diameter and spacing depend on your structural design. Google a bit and you can find good info on how to engineer it properly. Then screw post anchors into the footings, put in your posts, beams, joists and decking.
gardenweb.com has a porches & decks forum where you can get some good help on your specific project. Decks are definitely DIY-able, and great fun in the summer.