we performed human skin fibroblast isolation mincing the tissue in very small pieces, let them attach on plastiv for 24-48 hours with a very small medium volume (just covering a 35mm plate) and then we fill the dish in order to cover the culture with D-MEM 10% FBS. After 5-7 days you should see cells coming out from the skin pieces cultured.
Usually we use little pieces of skin, we let them attach in 35mm petri dish for 24-48 hrs (make sure the dermis is in direct contact with the culture surface). We use DMEM with 20% serum and change media every 2 days. Just pay attention not to move the culture dish too much, in order to avoid detachment of the tissue. In less than 2 weeks you should see fibroblasts coming out.
I used human fibroblast biopsy and cut it into 2 mm small pieces. But I guess, my fault was not to contact dermis and culture disk surface. The other things that I applied were the same as Cesare's suggestions (except for 30% of FBS). I used 30% of FBS on some suggestions in ResearchGate. Last I used 0,5 % CO2.