No heat, until driving it for 3 miles

Ok, So, first, if this answer to my question is really easy, forgive me, I'm just a girl, and as incredibly stupid as it sounds, I don't know what I'm doing!!!
About 3 weeks ago, I noticed that when I turned on my heat, it wouldn't come. So, I checked my coolant (yes, I can do that) and it was bone-dry. So, I filled it. And, it kind've worked again, however, it didn't kick in until I had been driving the car for about 3 miles. Then that night, I got back into my car after work, and no heat again. When I got home, I checked, and again, my coolant was dry. So, I filled that up again, and looked under my car, and I saw an almost steady stream of the yummy antifreeze creating a puddle under my car. I've been checking everyday since then, and usually, I would have to fill it every 2 days, or so. However, this is day 4 of me not filling it, and it's still full, I think it has a mind of its own. I bought a new water pump, as my mechanic said "it could be that," as he looked under my hood, and saw steam coming from where he said it was located. I haven't had the chance to get that done yet though. So, my car, while warming it up, isn't getting the heat I want it to, and doesn't usually warm up for a couple of miles. Can anyone tell me if it does sound like a water pump problem??? Or does it sound like something else?