One reason to juice the pomegranate rather than buy storebought for cooking is because when you pulverize the seeds, they impart a slight bitterness to the juice. Storebought juice tends to have a less bitter taste because it’s made with the drinking palate in mind. So, if you use storebought juice for cooking, you might want to cut the sweetness with lemon.
To de-seed a pomegranate, cut the fruit in half. Then, you can either submerge the halves in water and pull off the pith. The seeds will sink to the bottom of the bowl and the pith will float on top. Don’t strain, just collect the seeds out of the water. Alternatively, you can hold the pomegranate, cut side down, over a sieve (put the sieve over a bowl if you don’t want pomegranate juice to stain your countertops), grab a large wooden spoon, and using the back side of the spoon, hit the pomegranate firmly until the seeds fall out into your hand.
There are several methods for juicing pomegranates. Obviously, if you have a juicer or food mill, you can use those. If you don’t, you can gently pulse the seeds in a blender or small food processor in batches or use a mortar-and-pestle, and then strain out the seed fibers. And if you’re really strapped for kitchen equipment, a sturdy Ziploc bag and a nice bottle of wine also works. Just make sure the bag is strong enough so you don’t bash through it and get pomegranate juice (or wine) everywhere–double-bagging might be a good idea!