Which foods are high in umami ("savoriness") and how can I cook them to maximize the umami taste?
Umami comes from a very specific source: Glutamates and glutamic acid. MSG is a glutamate, so it is one of the best ways to add a bit of umami.
Meat, Kombu seaweed (used to make Dashi, and hence Dashi), mushrooms, onions, cheeses, soy and other beans, most high-protein foods.
Here is a quick list. Most are prepared foods, but it should give you an idea.
Marmite / Vegemite contain autolyzed yeast extract, which is a very high source of umami. These spreads are therefore also great for bringing out savory flavours.