Many wine tastings in the Bordeaux region are free. Find a place to stay (a B&B would probably be cheaper for you) in the Medoc and drive up and down the main road (D2) for tastings. Some of the bigger producers will have a small fee (Rothchild for example) but we tasted from lesser producers without any trouble.
Don't stay in the city of Bordeaux if you want a real wine country experience. Stay in the Medoc, St. Emilion, Sauterne...
We found "off the beaten path" places by reading Kim Millon's books about wine roads in France and Italy (they are rather old but the information is good). I'm sure there are other more current books - I just haven't done the research lately.