I have a 2003 automatic sunfire and my speedometer isn't working and sometimes it will jump all over the place and added to this problem my car will not change gears right and sometimes she changes hard. Also sometimes when I'm going 50 or 60km and let off the gas she will go back into first gear. I was wondering if this is happing because my speed sensor is give out?

That would be the most likely suspect

Yes, replace the Variable Speed Sensor (VSS) and clear all codes, if any, with a code reader. It's not unusual, in these situations, for ABS codes and DTCs to be thrown.

I think you may both be wrong.

GM has been plagued with instrument cluster problems for several years.
The problem stems from the tiny servo motors that vender's used to provide GM with clusters.

Recently, GM has been generous with replacing faulty clusters, even if the factory warranty has expired, either due to time or mileage.
Contact with your local dealer is the only way to find out if your problem can be addressed by this coverage.
Be patient and be polite. You'll tame more bee's with honey than you will with vinegar.
It's only a phone call.