If the car has a manual gearbox, check that the clutch operates smoothly and all gears engage easily.
If the gear change stiffens as revs increase the clutch may be worn. Check for clutch slip by driving the car up a hill in top gear.
When driving change down into each gear from a higher speed than normal to test the synchromesh. If the gears baulk or crunch, or if the gearbox whines excessively, gearbox overhaul or replacement are the only solutions.
If the car has automatic transmission check the transmission dipstick for correct fluid level, this is best done with the engine hot and idling.
Also smell the dipstick, if it smells burnt steer clear of the car!
When driving check that the transmission changes down into each gear properly under full acceleration and at the right time.