Thermostat or a water pump?

Hi so I'm going to try and make this as clear and straight forward as possible. So my car started overheating and I didn't know why eventually it led to my radiator cracking. I just replaced the radiator yesterday and I test drove it to see if it's still overheating. It was fine driving around the neighborhood but that changed when I took it to the road. It went hot. I'm glad there weren't any old people to get in my way before it blew. I filled it with fresh coolant but when I got home I checked it and my reservoir was empty. My car doesn't have a radiator cap. I did notice before that when I would drive it would start to overheat but would settle down while I drove but would go back up when I stop at a red light or something. When I test drove it, it just went into the HOT but didn't go down while I was driving. It did the other day before I replaced the radiator. I drive a 2001 Ford Escape XLT V6 3.0L 2WD. So could this be a bad water pump or a bad thermostat? I would like to know before I just start buying stuff. There's a difference between the prices. I know you can't do as much without seeing it but my car can't go anywhere because of the overheating. What do you think?

The first problem is no radiator cap and you're loosing coolant. The cooling system must have a cap. Try putting a cap on it before you go to the time and expense to check the thermostat and water pump.

Check the coolant level by slowly adding coolant until it takes no more, let the engine cool off and do that again. Check the fan relay.- replace it…$15 part. If the fan runs with the ac on, the fan is good, but if it won't turn on while sitting still and engine is hot, the relay is bad.