When scientists entangle photons or atoms or other objects, is it only their state that gets entangled? What about their proximity in space? If you move one sideways will you see the other one move sideways as well?
In general, entanglement of two objects is not necessarily connected to spatial proximity. Entanglement means that you need to consider both particles' quantum states when you want to describe the properties of the system. You cannot describe an entangled particle's quantum state independently of the other one.
There is no restriction as to which properties can be correlated. It could be spin, polarization, or even position. But even if the particles are entangled with respect to their position - that does not mean that you can control one by manipulating the other. It means that if you measure the position of one particle (i.e. "look at it"), the position of the other one will be known as well. At this point, the entangled state brakes down. The particles are no longer entangled when you perform the measurement and the positions are known.
So to wrap it up, let's move through your questions: Yes, it is their states that are entangled. Spatial position can be correlated, too. No, you will not see one particle move because of the other due to entanglement.