Replaced the battery in a 2001 Volkswagen Cabrio... and now it won't idle! Sometimes it won't even start. Do I need to take it to the dealership or can this be fixed easily?

In the link

https://www.google.com/search?q=2001+Volkswagen+Cabrio+wiring+diagram

..

notify or visit dealer have id and proof of ownership they usually use the vin number to retrieve codes if dealership does the reprogramming they sometimes charge you but usually dont to give you the code

check the voltage regulator

Have you had the flywheel ring gear checked out for worn or broken teeth this can can stop the engine turning over

replacing battery more likely erase ECM memory.best to start vechicle let enginet idle until ECM relearn memory.your vechicle has to complete full drive cycle before ECM MEMORY return if not let dealership reset engine idle.

you will have to go to the dealer to get the radio code as these are anti-theft radios so if they are stolen they won`t work without the code.

My first guess would be a bad idle air control valve if an inspection shows nothing apparent.

Take out the radio. sometimes the code has been written on it. It also might be written on the owners manual.
If not, take the radio and your VIN (vehicle identification number) to the dealer with proof of ownership of the car.
If you are nice to them they just might give you the code. My local dealership gave my code to me like this.

When my 1999 Cabrio was new the cruise control went out. It took the dealership and along with a VW expert 10 days to find a pinched harness assembly under the battery box.

I had the exact problem with my wifes 99 Cabrio - parked for 9 months. Ended up having to have the shop reprogram the ECU - it then worked fine.