Just read it how you feel like it.
家 is usually read as いえ, but it’s read as うち mainly when the speaker is talking about his home in an informal matter. But usually うち is written in kana, so just stick with いえ.
For 明日, あした is more popular, あす is a little bit more formal. There’s also みょうにち which is probably the most formal. Again, just go with the most used あした, but if you feel like there’s a sense of formality, go with one of the other two.
Honestly, the only time I see あす is on the news, usually when they’re talking about weather (at least for the news in Nara). I always see it written in kana. However, I do hear あす and あした used interchangeably, so it just depends on the speaker. I never really hear あす all that much when I lived in Tokyo, but now that I live Nara, I hear people use it a lot more. But it just depends on the speaker I think.
So, if you see 明日, I would assume the reading あした. But at least where I live, both あす and あした are used in conversation, so it’s good to know both.
As others have mentioned, うち usually refers to the speakers own household. For example, if my coworker says, うちの息子(むすこ = son), she is saying, “My son” (you could interpret this as"my household’s son" if it makes remembering the distinction easier). There are situations where うちの means “mine” or “our” (kind of like 自分の), so you can think of うち being something more personal or close (to your house I guess), vs いえ(家) which is just a basic term for a house. And again, as other’s have said, うち is very likely to be written in kana, while 家 will be いえ.