2003 Toyota Corolla 72,000 miles

My thermometer after driving for a while goes to “H” hot and then eventually the check engine light comes on (I just had the radiator replace) and need new tires....
Does this sound like something serious? Someone asked me should I just get a new car but I only have 72,000 miles on it pretty much only drive locally and have stayed on top of oil changes etc? thoughts please and thanks....
Since you have already replaced your radiator, its always a good idea to have your cooling system/radiator cap pressure tested for leaks and to ensure the cap is holding the specified pressure. In addition, as general practice you should also replace your thermostat whenever your radiator is replaced. This is cheap insurance since parts are fairly cheap. I typically refresh the entire cooling system @75K miles with a new water pump, thermostat, hoses, etc. It has been my experience that radiators (foreign & domestic) with plastic tanks, will begin to leak at around 75K miles anyways.

If you can resolve the issue without a major expense, it should go to 150K miles easily.
Your car is just broken in. Our 2007 CE has around 342K. We did replace the clutch around a quarter million miles (my wife learned to drive a stick on this car), and we swapped in a used transaxle around 300K when I heard the transmission output/differential input bearing growling. Our 2007 S has around 172K miles.

If you have a good radiator and enough coolant, yet are still overheating, something is preventing adequate circulation of that coolant. Water pumps very seldom die. I have seen thermostats fail shut. On all but a very few cars, the “get you home” field repair is to remove the thermostat; the car just runs cool. Swap in a new thermostat. They are not expensive. Get a good one. I sometimes satisfy my curiosity by putting the thermostat in a pot of water and gradually heating it while I watch a thermometer. I can see if my old one was bad, and if my new one opens at the specified temperature.

On a very few cars (mid 80’s Civic Si) you must get the factory thermostat, the others won’t work. I have not heard this about the engine in our Corollas.