Have a 2000 INTRGUE that show to be running hot

Have a 2000 INTRGUE that show to be running hot. She ( mother in law) had the Thermastat replaced. It take it a long time to get hot at idle. But I can drive it down the road for about a mile and the temp guage goes up to almost in the red. Push it up in to N and run the RPMs up to above 2000 to 3000 and the temp comes back down. And then starts to go back up. Also when it goes back up. There is like a poping noise. I can only here it from the inside of the car. Out side with the hood up and the car runing I can not hear it. But stick my head back in side and I can hear it. Also when I got back from road test and the gauge showing it to be hot. I can touch the hoses, any part if the ENG. It does not seem tht hot.
Any help? If I can get some good help I will be more than happy to give a donation. After I get some help. I did this before and no one ever gave me any thing. But I lost $50.00