this is hands down the BEST way to fill a cooling system. No burping required, and you've completely filled the cooling system in a couple minutes as opposed to letting it idle with the heater on, etc while bubbles purge out for a half hour...
https://www.amazon.com/dp/ (commissions earned)
I've use it for many years and can't imagine filling a cooling system without it. If there's a leak in the system, it will find it too. It basically pulls the entire cooling system into a vacuum. The hoses collapse, and there's no air left. Flip the valve with the hose in a jug of fresh coolant and it sucks the coolant right into the system. Done. Because it pulls the system into a vacuum, if there's a leak you'll see it losing vacuum when it's all pulled down, instead of holding steady on the gauge.
Otherwise, the only way to do it right is to get one of these
https://www.amazon.com/dp/ (commissions earned)
it's a little cheaper, but you have to let the car idle for a half hour, revving it occasionally, with the heater turned on so coolant circulates through the heater core, etc. and just keep going until no more bubbles come out. can be a hassle, especially for some cars like a Toyota MR2 or Toyota Previa, where the engine is in the back of the car, or some Fords are a nightmare as well and CAN'T be done without an airlift...
check to make sure the radiator cap is in good condition as well. How is the coolant level, it is maintaining "full" or is it losing coolant? I've seen guys leave the radiator petcock a little loose as well, and it will drip out under pressure but not "leak" when you're sitting so you never know it's leaking anything. If it's maintaining the full level you may just have an air pocket where the coolant temp sensor is, so it reads a lot hotter than it really is...
lot of possibilities. but I agree first step I would do is drain the coolant and refill again, taking time and care to make sure there's no air pockets.