Q:

Do things have colors because their electrons are getting excited when photons hit them?

Atomic electron transitions can be caused by absorbing a photon with a certain wavelength. An electron jumps to an higher energy level, then it falls back and a photon is emitted. The perceived color of the photon depends on the energy absorbed by the electron.
Could we say that electrons in the atoms of different objects are excited when white light hits them, and they release photons which in turn causes the object have a color?

A:

Could we say that electrons in the atoms of different objects are excited when white light hits them, and they release photons which in turn causes the object have a color?

What you describe there is known as fluorescence but that's not what happens when white light hits a coloured object, at least not in most cases.
Most objects are made up of molecules. These molecules are in turn made up of atoms, bound together by so-called molecular orbitals (MO, aka bonds).
When a photon of the right wavelength (and thus right energy) hits an electron in an MO the photon is absorbed and the electron is moved to a state of higher energy. 
The white light, minus some of the absorbed photons, causes the reflected (or transmitted, in the case of transparent objects) light now to appear coloured.
So the phenomenon is caused by VIS photon absorption, not fluorescence.

