How can my wife and I balance our time together when we have our own apartment but have families to take care of who need us almost every day?
You didn't mention what part of the world you are in. I suspect it may not be North America.
You and your wife will each need to talk to your families. You need to let them know that you care about them, and want to help them, but that you can not be with them every day.
Even in cultures where parents must be obeyed and respected, cared for and suported, no one can expect that to happen every single day.
I would suggest that you ask them to make a llist of what they need help with and that you will be over on Tuesday, Thursday and Sunday, or something like that to help them. Your wife should say the same to her parents.
If you start this way, and both stick with only these days, you will each be with your parents on the same days. This will leave you with four days where you and your wife will be home together. After a while, you will find your parents do more things for themselves, or find someone else to help.
You are stil being caring and dutiful.
Soon, you can both tell your parents you can only come on Wednesday and Sunday, then later still, only on the weekend.
I am guessing neither of you wants to be firm, or rude, or hurtful to your parents, but your parents must begin to realize that you and your wife are a new family.
Talk to your parents, work out a schedule, set some boundaries and stick to them.
You and your wife deserve time to build your life together.