There are a number of tests that can be performed find the cause. The PCM ( powertrain control module)should be scanned for trouble codes. This transmission iselectronically controlled so it might just be a bad connection or a solenoid malfunctioning. This is a complicated procedure and without a transmission service manual and proper testing equipment it is impossible to diagnose.
Replace your transaxle
More likely it has something to do with pressure problems in your transmission, check the fluid and replace the transmission filter. If the problem is not fixed check the vaccum lines running to the transmission and the vaccum off the engine.
A seized up transmission can cause a Volkswagen Golf with an automatic transmission to not shift and racing RMPs after second. A worn gear can also prevent shifting.
Quite possibly a worn second gear clutch pack.
That would be the second gear of your automatic transmission.
An automatic transmission might be slow shifting from first to second because of low transmission fluid. You might also have damaged the gears in the transmission at the area between first and second.
See if you can locate the cable that goes to the lever on the front of the transmission and determine if that cable is broken. If you find a broken cable, replace it, if not you may have a serious transmission problem.
There are several causes that an automatic transmission will not want to shift into second gear. The two most common causes are transmission pump issues and trouble with the vacuum modulator.
Do a VIN search and the VIN will tell you the original options.
The first thing I would do is change the transmission filter and fluid and have the transmission flushed. This is the cheapest thing you can do to remedy the problem, and it should be done every few years. I hope this helps you. Mark
Had to replace mine twice, at 30K miles (under warranty) and 70K. Second cost me $2800 w/ three year warranty. Was done at a local shop, not a chain.
There could be two reasons for this. The first is the transmission is completely out and the second is the linkage in the transmission is broken.