If you've looked through both lessons 6 and 3 you may have noticed that while a family is a 'teaghlach,' a house is a 'teach.' This isn't by accident, either. Teaghlach literally means household. However, the definition isn't limited to just your immediate family. It can be used for your extended family as well.
If you strictly wanted to refer to only your household, though, you could use the phrase líon tí, which literally translates to a house's count. That tí is actually a the genitive form of the word teach, but we won't worry about those kinds of changes until later.
Another Irish word that you're probably familiar with is clann. Because of its use in English as a synonym to tribe, you might think that it refers to a whole family. However, that's not the case. In fact, it refers specifically to a couple's children. So, if you wanted to say 'my children,' you would say mo chlann.
Ireland has a very family focused culture, so it has plenty of different words for family members!