Reading labels and looking for ingredients described by a number is a good idea, but maybe take things slow so it's not so overwhelming at first.
For instance, concentrate on finding a good quality source for fruit and veg - especially if it's certified organic. Look for things that are in season and maybe check out local farmer's markets so you have a good idea that it's probaby fresh and hasn't been stored for ages.

Then, when you have that down, you can start with dairy, or whatever. Many grated cheeses have preservative added, while the blocks don't, so it's better to buy the block and grate it yourself, for example.
Then move on to breads/canned things/jar sauces etc.

Generally speaking, the closer something is to how it is naturally, the less junk is probably added. eg. real butter would be a better choice than margarine.
Choose products with real sugar over diet alternatives.

Asian recipes have heaps of flavour using natural ingredients (stir fry for example) and are really quick and easy - just stay away from pre-made sauces.

It's sad that you probably do pay more for quality food, but not always. It might seem hard at first but what an awesome thing you're doing for your whole family! Good luck