2000 Suzuki esteem wagon, rpms race when car is rolling

Engine idle speed is limited by reducing the amount of air entering the engine. When an accelerator pedal is pressed, it pulls a cable which opens a throttle plate which then allows more air to enter causing engine RPMs to increase. At an idle, the throttle plate should be closed as to not increase the RPM. There's also a part called the idle air control (IAC) valve which controls how much air bypasses the closed throttle plate at an idle. Lastly, air could enter the engine via an intake gasket or vacuum hose leak; the same rules apply, the split hose or gasket allows air to enter the engine and RPMs will rise. To summarize, a throttle plate not closing, and IAC valve stuck open, or a split intake gasket or vacuum hose could all cause a higher than normal idle condition. You can read more about how to troubleshoot high idle conditions here: https://www.yourmechanic.com/article/how-to-troubleshoot-a-car-idling-at-a-high-speed-by-tim-charlet

check ICS for correct operation
engine idle speed is 850 -950 rpms for auto and 750-850 for manual
at 1200 rpms you are not on the idle circuits but on power circuits
run the fault codes

Sounds like 2nd gear in the transmission is slipping. Cars usually shift at 15-20 mph from first to 2nd. First check the trans fluid.

While it is idling do you have your heater on or defrost on? If so the compressor will kick on making the car move some and the idle to increase because it is under load from the compressor kicking on. This is normal Hope this helps

I would start with ect. That is the engine coolant temperature (sensor). That sensor monitors the coolant temp and tells the computer to increase engine RPM so the engine warms up faster. In the mornings when you first start your car and it idols high, that sensor is sending that information. If you have a code for this sensor start there.

COULD be the TPS sensor or could be more serious

your vehicle will need a complete computer diagnostics to determine this problem and its solution. there are a lot of unknown variables to be able to accurately begin to solve this.

make sure the coolant level is all the way full

sound very much like a vacuum leak.... also maybe your timing is off? possibly advanced? try retarding it with the distributor if thats the case...

This happened to my 2001 Regal... There is a factory recall on the Catalytic Converter going bad. I had mine replaced with the recall and haven't had a problem since