Step 1: If the meat pot contained actual meat and was still very hot when poured into the cold sink, then yes the sink becomes meat. However, if there were only pareve foods in the meat pot or of the contents of the pot were less than 110F the sink would retain its original status.
Step 2: If the milk pot and the sink were both dry and clean, then both the sink and the pot are permitted and nothing more needs to be done because flavor id not imparted from one utensil (sink) to another (pot, or visa versa) without an intermediary. However, if the pot had milk on the outside or if the sink had meat residue, then both the sink and pot are now treif and need to be kashered. A small amount of moisture on the pot or in the sink is not a problem.
Step 3: If kashering is required, the pot should not be used for 24 hours after a thorough cleaning and then must be immersed in a large vat of boiling water. There are many other important details, and I don't recommend doing this yourself if you don't have prior experience. Regarding the sink, if it is made from unpainted stainless steel it can be kashered as for Pesach. However, if it is made from porcelain or enamel it is a much bigger problem.