Taxes for owning two homes

Here's a question. I'm thinking of buying an inexpensive second home in a state that has no state income tax.

Can I claim my residence (and taxes) in only one state of will my original state look for their cut?

If I cannot claim only one residence than I am also considering gifting my original property to my daughter and purchasing a second home for my personal residence. What do you guys think? Any drawbacks to owning two residencies?