I usually ask my kids to peel the garlic and ginger while they are watching tv :) That's the hardest part! I store it in the fridge. Don't be alarmed if it starts to turn greenish ...it usually discolours within an hour or so. It's just fine :)
Put both together in a food processor or blender and chop until it's as fine as it will go (add a bit of water to it... not too much... just enough so that it's not hard on the blender!) Put into a glass jar and keep in the fridge.
Use in Indian/Thai/Chinese recipes or whenever ginger-garlic paste is called for.