Doing scientific experiments is a lot like cooking, we use recipes, called protocols. A large part of scientific research is dedicated to the development of new protocols or improvement of old ones to measure a larger number of samples more efficiently and with a higher sensitivity.
Here we will explain how we purify protein, purify DNA, cut and paste DNA for cloning, and modify the DNA sequence by adding mutations.
DNA-protein complexes are so small we cannot even see them with a microscope, but there are a few techniques we can use to measure them. We can separate DNA-protein complexes by gel electrophoresis, visualize the protein binding sites on the DNA by footprinting experiments, or use Atomic Force Microscopy, which is the only way to actually "see" a complex in its context. Finally Cryo Electron Microscopy in recent years has become a very powerful technique to characterize the structure of multiprotein complexes and protein-DNA interaction.
Try purifying DNA at home! (here's a web page where they explain how)