The question of whether to remove collard stems from the leaves and discard them or cook them separately is largely a matter of convention and personal taste. The stems are perfectly edible, but if you like them soft, they will take longer to cook than the leaves, and they should therefore be cooked separately. However, if you enjoy your vegetables with plenty of texture, you can save a step in your cooking process by simply leaving the stems in your collard greens and cooking them all together.
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If you find collard stems too tough to eat or if you simply prefer them cooked longer than the leaves, you can remove them and either discard them or cook them longer than the leaves. You can remove the stem from a collard leaf by grasping the stem with one hand and pulling the leaf away with the other hand. Alternately, you can cut the leaf away from the stem by cutting a thin "V" and separating it from the leaf.
If you're not going to enjoy collard stems no matter how thoroughly they are cooked, you should discard them. If you don't want to throw them away, you can compost them, or if you have chickens or goats, they'll be happy to eat your collard stems.
Cooking Stems Separately
If you like eating collard stems but you like them to be softer than they can get in the time it takes to cook the leaves, you can cook them separately or at least as a separate step. Chop them into small pieces, about 1/2 inch to 1 inch long. The smaller you cut them, the less time they will take to cook. Start them early in the cooking process, along with longer cooking ingredients such as onions and root vegetables. Add the leaves to the dish later in the cooking process after the stems are softened to your satisfaction.
Cooking Leaves and Stems Together
Unlike the stems of tougher greens such as kale, collard stems do get soft and many people find them appealing even when they are not completely tender. If you like or don't mind your collard stems a bit chewy, you can save yourself some time and work by simply leaving them in the leaves and cooking both parts together. Divide the collards in strips of 1/2 to 1 inch, cutting perpendicular to the stem. Stew, steam or saute the strips until both the stems and the leaves are cooked to your satisfaction.