Recombinant DNA technology involves isolating DNA from organisms, cutting it with restriction enzymes, ligating the cut DNA fragments into vectors, inserting the recombinant DNA into host cells, culturing the host cells to produce multiple copies, and extracting the desired product. Key steps include isolating pure DNA, cutting source and vector DNA with the same enzyme, ligating the gene of interest into the vector, amplifying the gene using PCR, transforming host cells to incorporate the recombinant DNA, and using selective markers like antibiotic resistance to identify transformed cells.