I own a house and some family members will soon be living with me. If I charge them rent, how will my taxes and their taxes be affected?

You would have income to report on your taxes. You might be able to claim depreciation but that can be problematic by itself.

You might consider having them pay some utility bill or something rather than cash directly to you.

You first need to have a rental agreement, and money ( for when it all goes bad and you need to evict them) Next you will have expenses with them there, and while that rent is income, if you don't make it too high, your expeneses should off set any of the rent.

Their taxes are not effected at all, since rent is not deductable

First, I agree with FR Chuck; GET a rental agreement!

Now, if they are living with you IN YOUR HOME, you can charge them rent and it will NOT affect your taxes. The IRS is not interested in trying to track rent for renting a room to a family member. The reason is that normally, after you factor in pro-rated utility costs and pro-rated depreciation, the Schedule E produces a bottom-line LOSS. However, you CANNOT claim a loss on rental property that is PART of your primary home. THAT is why the IRS does not even want you to report it.

If you are renting a separate house to them, you must treat them like you would a perfect stranger, or the IRS will take a HARD look at your rental arrangement.