I think it boils down to ... which do you have ... time or money.
If you have more money than time, you can have the old dirt removed, a drainage system installed and fresh fertile topsoil brought in.
Ok, so you aren't bill gates. A more cost effective solution is to ammend the soil you currently have. First you must decide if raising the level of the garden a few inches is going to be a problem. If so, you have to remove some of the dirt. Most of us don't mind it being higher so just start ammending.
All of the previous posts have been good. You can buy bags of amendments at the box store, but that gets expensive fast. One dollar for a 1 cu ft bag of compost works out to 27 bucks a yard. You can buy precomposted wood and organic compost by the truck load from many suppliers. Prices in the 18-35/yd delivered are typical. 2-3 inches on top of the garden then turned in will do wonders.
I am not a fan of adding sand to clay. I feel that you wind up with sandy clay. Organics are the key to breaking down clay.
I have had mixed feelings about drywall. It sure is cheap if you let it be known that you take drywall scraps. You will soon get more than you need. A layer of drywall on the garden while the wood chips mixed with grass composts will get you going pretty good in about a years time. Yes, the wood chips will ... what's the word ... absorb nitrogen ... as they compost, but this nitrogen will be available later in the cycle.
The one thing I do think needs to be said. Don't worry if you don't get it right within 48 hours. Turning raw clay into rich fertile soil can take years, and any mistakes you make at the beginning can usually be corrected. That is, with the possible exception of adding sand which would be a real pain to remove. I would suggest you do your own research and make up your own mind about whether you want to add sand or not.