One must wait a significant amount of time between eating meat and dairy. Opinions differ, and vary from one or two to six hours. This is because fatty residues and meat particles tend to cling to the mouth. From dairy to meat, however, one need only rinse one's mouth and eat a neutral solid like bread, unless the dairy product in question is also of a type that tends to stick in the mouth. For example, most margarines are dairy for kosher purposes, because they contain a small quantity of whey or other dairy products to give it a dairy-like taste.
Animal fat is considered meat for purposes of kashrut.
You should read the ingredients very carefully, even if the product is kosher-certified. Thus, if you cook chicken soup in a saucepan, the pan becomes meat. If you thereafter use the same saucepan to heat up some warm milk, the fleishig status of the pan is transmitted to the milk, and the milchig status of the milk is transmitted to the pan, making both the pan and the milk a forbidden mixture. Kosher status can be transmitted from the food to the utensil or from the utensil to the food only in the presence of heat, thus if you are eating cold food in a non-kosher establishment, the condition of the plates is not an issue. Likewise, you could use the same knife to slice cold cuts and cheese, as long as you clean it in between, but this is not really a recommended procedure, because it increases the likelihood of mistakes.