Antifreeze

Try a new radiator cap and check the hose in between the reservoir tank and radiator for a leak and/or loose clamp on either end.

I think that either of those is the problem.
The coolant should go back into the radiator from the overflow tank when the engine cools and coolant shrinks. If there is a hole or leak in that hose it will just suck in air and not coolant.
Or if the cap is bad it would prevent the coolant from getting sucked back in.
Unless the problem is just the fact that the coolant is boiling over anyhow, if that was the case then perhaps a new thermostat.

Yes, that is what it sounds like, either a clogged up radiator core tubes or the water pump is not circulating the coolant.
You can usually tell if and where the radiator is clogged. Once the engine gets warm and the thermostat opens up, shut off the engine and run your hand over the entire surface of that radiator core, if you feel any cool or warm spots that is where the clog is.
If this is a cross flow radiator meaning that if you leave the radiator cap off, once the engine begins to warm up and the thermostat opens, you should see the coolant begin to flow out the top tubes looking down into the radiator vent hole. If the flow is not nice and strong once the thermostat opens up and them top tubes are not clogged up with rust, you would usually be able to see rust covering the ends of those top tubes, then if the flow is not good enough, perhaps it is the water pump. But first, make sure also that the bottom hose is not weak or missing the metal spring they sometimes stick inside those bottom hoses and if it is weak, it might suck closed when the engine is running, so first when cold, squeeze that hose and feel it, is it soft? Can you squeeze it closed real easy? If so, you may want to replace it and flush the entire cooling system if there appears to be any rust in the coolant.
A radiator can be rodded out if it is too expensive for a new one. What they do, is unsolder one tank then stick a long thin metal rod down every single tube cleaning it out. Then solder the tank back on and check for leaks.