Many people rely on good old-fashioned vanilla extract for their baking needs largely because it's easy to find and relatively affordable. Vanilla extract is made by macerating vanilla beans in a mixture of alcohol and water. When purchasing a bottle, avoid the imitation versions as they they tend to have a weaker, often tinny aftertaste. Splurge for the real stuff here.
Speaking of splurges, vanilla beans, while on the pricier side, are the ultimate in flavoring and scenting baked goods. Vanilla beans consist of an almost waxy dark brown pod filled with thousands of little brown flavorful specks. When purchasing them, you want to make sure the beans are plump and smooth, never dry. They should smell highly fragrant and have a slight shine to them. The big perk to using vanilla beans is the way the thousands of little black dots fleck throughout your batter. For lighter, fluffier desserts like white cakes, cupcakes, or muffins I'll always use a vanilla bean because the flavor is so intense and the visual component of being able to see the little black specks in the finished product is always satisfying.
What about vanilla paste? In many ways, vanilla paste could be thought of as a great in between option. It's essentially a small jar of the scraped-out vanilla pod, so you're going to get that super fragrant, sweet, speckled end product with the convenience of a quick scoop of the teaspoon. I don't find it all that much work to select, store, and use vanilla beans but some folks are less intimidated by the paste. I say whatever works and gets you excited about baking!
As far as conversions go, 1 teaspoon of vanilla extract is equal to one 2-inch piece of vanilla bean, so 1 typical vanilla bean will equal 3 teaspoons extract. As for vanilla paste, there are usually directions on each jar indicating proper equivalencies, so consult that first as the concentrations can differ.