First off, do some research. Several of the forum members have very informative Web sites listing all the hydroponic applications with pictures and tutorials. Just click on the person's profile link for that information.

The most basic system would be a flood and drain. Place it in a south-facing window (if you live in the northern hemisphere!) and that's about as easy as it gets without having to buy a lot of stuff.

I feel the most basic is like the one you have experience with. You just setup a reservoir and place plants on top, letting roots dangle in the nute solution. An air stone in the reservoir will keep the mix aerated. Leave room to be able to get into the reservoir so you can occasionally mix it up thoroughly and test the water. Use a cheap pH test kit (the drops). I suggest an EC meter, which measures how much nutrient is in the water to put it simply. That will cost you a bit of money. I would also stick with commercial nutes rather than making your own until you get used to hydro. Then you might have the funds, time, and room to do your own experimenting. Use sunlight if you can.

Most advice can only be given after certain info is given by you. That's one of the many values of hydro. You can adapt your system to your available environment. Describe the environment you plan to grow in.

Area and height of the space?
Indoor or outdoor?
Sunlight availability?
Ventilation?
Temperature?
Plants you plan to grow?