Your engine will speed up & the car won't go any faster (transmission slipping) Or you can pull the dipstick & smell the transmission fluid if it smells burnt you may have a problem with the transmission.
when going up a hill you can really notice the gears are jumping when changing. Also the car is also making a grinding sound. Is this the transmission?
If it's difficult to switch gears, if there's unnatural noises when switching gears, if the engine runs too high or too low when accelerating/decelerating ... list goes...