Ok, so recently moved into an older model one story home. In the kitchen, when I squeeze the sprayer, I don't have full pressure and the faucet still runs water. I know in my past apartments, when I would squeeze the sprayer, I would have funn pressure and no water would come out of the faucet.

What can I do to stop water from coming out of the faucet when I use the sprayer?

The first thing to do is pull out the hose and unscrew the sprayer. Turn on the water with the hose pointing into the drain. Lots of water? If so, the sprayer valve is clogged or bad. Cheap replacement at HD or Lowe's. If no or little water, the supply hose could be kinked or plugged or the diverter valve is plugged or bad. Check this out second. Finally, depending on the faucet, you need to remove the spout and you will see a black or white circular small button recessed in the body of the faucet. Remove the valve, flush it out and soak the diverter in vineger to remove mineral deposits. Clean out the hole where if fits, reassemble and it should work. You can also Google "cleaning a hose sprayer diverter valve" for more help.

Before I can answer I need more information. What's the house pressure? If it's low that would give you the systems that you have.
What's the brand name of your faucet? One handle or two knobs? How old is it? If you don't have low pressure then I would suspect a faulty diverter or one that has mineral buildup blocking the action. Back to you, Tom