Car is overheating

So I bought this car and it was running fine with the exception of the noise coming from the engine kind of sounding like the power steering was going out, or a bad blower motor. The noise would be intermittent until the car warmed up. It would not be present when I turned the steering wheel. But the other day I drove it and it started overheating, so I checked the antifreeze and it needed some so I put some in. I started the car and drove it back home with it going back and forth from overheating to normal. It would blow cold air then hot air with gauge going up and down. Got it home and left it overnight. The next day I got in my car and started it, and low and behold it didn't overheat during the time I drove my car. Then next day got in my car and it's cold and I had to clear some snow off the vehicle so I turned the car on to let it warm up and got in and drove to my destination by the time I got there which was only 2 miles away it was overheating again blowing cold air out the vents. I let it sit for 10 minutes and got in to drive home and it started to want to overheat but then the heat kicked on and the temperature went down enough to make it home without it overheating. I don't know what to do or what to think is wrong with it. I mean it was running and heating up just fine now all the sudden its starting to overheat. I don't have a lot of money to fix and it's my daily commute car so please what do you think it is. How can I check if it's the water pump?