Making Tomato Sauce from Tomato Paste
I can give you the Italian answer - first of all, normally we don't use tomato paste to make sauce, but rather to add a tomato "kick" to recipes. Tomato paste is simply tomato puree that has been cooked down to a high degree of concentration. 

A basic tomato sauce is made by 


making a soffritto with onion, carrot and celery (plus other flavors)
adding tomato puree, or "pelati", more rarely fresh tomato
cooking the sauce down until the taste and thickness is what you want


If you want to use paste instead of puree, the third step has to be omitted or greatly reduced in duration. Keep in mind that tomato paste has its own taste, and that taste will remain in the final sauce.