Did the block get purged? Does sound like an air issue. You can't leave the radiator cap off, and even if you did, on a vehicle that the cap is the highest point, they don't need the cap off anyway, that shows a lack of understanding of how the cap works. You fill through the overflow/reservoir, not the cap. There is a reason there is a hose from the cooling system to the overflow.

Sounds typically like a water pump problem.

Sounds like you have a lot of air in the cooling system. Take the cap off, let it idle to operating temperature. wait for the bubbling to subside. top off as required.

Take it to a mechanic and get it fixed ASAP, you could destroy your engine... You're extremely lucky it hasn't already happened.

Could be the water pump

I have a 2003 Hyundai Elantra and the radiator blew a week ago...replaced the radiator and it still lost coolant but I couldn't see from where...I replaced the thermostat for the hell of it and the problem continued...I finally noticed the coolant was coming out of the overflow tank so today I replaced the radiator cap, and I still have the same problem. But there is also no pressure in the system while this happens and the heat comes and goes...any help would be appreciated.