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The transmission may need a flush and filter, or in my 92 accord I solved the problem merely by changing the transmissions computer.
In automatic transmissions the second to third syncronizers always seem to go bad first, the syncronizers are worn, bad or gone. This is common in these transmissions. Also you might check your trans fluid level, sometimes a low level will cause the transmission not to shift correctly.
Are you sure that it's third it's not shifting into, and not fourth? The transmission is designed to not permit shifts into fourth (overdrive) until the engine and transmission have warmed up a bit.
A car with an automatic transmission shifts itself from 1st to 2nd to 3rd when you accelerate. You don't need to shift it...that's why they call it automatic.
Automatic transmissions work on a combination of fluid pressures and electronic signals. Make sure the fluid level is proper, and does not smell burnt. Burnt fluid can be a signal that discs are slipping in the transmission. Also check to see that the linkage is tight and set properly. An unadjusted linkage can cause the transmission to not go into the proper gear.
Your transmission's filter may be clogged, you may be low on transmission fluid, or the transmission may need repair or replacement. If the transmission was full of fluid, I would suggest a transmission flush, or replacing the transmission filter and fluid. If this doesn't help the transmission may actually be damaged and may need repair or replacement.
If your transmission is not shifting into third gear and overdrive it could be a parking pawl. This prevents the transmission from rotating.
There are tiny bearings and gears that make the trans shift smoothly and keep it in the right gear and when they go bad your car can develop a rough shift slipage and even trans failure
More than likely your shift cables need to be lubricated.
Check your tranny fluid. If it's full, you need to get your transmission checked. The screen could be dirty and need to be replaced, or the transmission itself could need to be repaired/replaced.
If this is an automatic transmission, which I am assuming it is it will not work the same as a standard. Only drive it in overdrive or 1st gear.
Could be low fluid,a bad sensor or transmission going bad.