Grammar

"the bread my wife makes is much better than that which we can buy at a store."
in the above sentence, can we leave out the word that or which?

You can, though the sentence is clearer with the words 'that which' included. You are making a comparison. To do so properly, you need the two subjects you are comparing: 'the bread my wife makes' AND '(the bread) we can buy at a store'. That which is essentially a replacement for the bread.

Fortunately, though the sentence you provided is correct, very few (young) people would use it. Instead, we would say "the bread my wife makes is much better than what we can buy at a store". It's less formal/proper, but much more common. Or, you could simply use: "the bread my wife makes is much better than the bread we can buy at a store".