I have wireless home phone AT&T Home Base. How do I get call forwarding to work?

What happens when you try?  Have you tried the in-forward to see if you get a confirmation tone on it? Looks pretty straight forward in the manual.

Call Forwarding To forward all calls, dial: (1) *21* (2) The 10-digit number to which you wish to forward your calls (3) # key (Example: *21*1234567890# forwards your calls to the phone number 123-456-7890) Please wait at least three (3) seconds and you hear a confirmation tone for the forwarding to complete before hanging up. To un-forward calls, dial #21#, please wait at least three (3) seconds and you hear a confirmation tone for the un-forwarding to complete before hanging up.

What do you hear when you enter the information?  There's all kinds of "doesn't work".  Is it just nothing silence? Are there tones or beeps?  Is there a busy signal?  What happens when you dial the cancel forward codes? 

​I get an immediate 3 beep tone then silence and after 15-20 seconds the "rapid busy" signal.  After than, the calls that were supposed to be forwarded go into voice mail after the caller enters the telephone number again.  Going to the #21# then turns off
the voice mail.  No Call Forwarding, only what I already have to take messages.

So it is forwarding, but to the voicemail and not the number you entered.  By any chance are you putting a 1 before the number?  As far as voicemail not working after dialing the code to remove forwarding, try going through the voicemail set up again to see if that corrects it. You may have to call tech support as your problem can't be found searching for it and if it were widespread there'd be more reports. 

​I do not put a "1" before the 10 digit number.  As to contacting Tech Support, AT&T does not tell me how to get to them.  All I can do is call Customer Support (an extremely funny name given how hard they are to get to and how little they support).

So what's the direct line to Tech Support?

I've never had to use AT&T tech support, but I've used others.  I simply asked for tech support when the phone was answered, if that doesn't work keep explaining this is a technical issue that can only be resolve by their tech support. You could also try reaching out to a board admin if that doesn't work.  

Your problem per your examples is you activate call forwarding and all it does is forward calls to voicemail no matter what number you enter

You deactivate call forwarding and voicemail quits working, proving it's forwarding calls to the voicemail

Since this is a customer to customer help forum I really think you need to push, escalate, or whatever else is available to get your problem resolved.  Please let us know what is found as it could help someone else in the future. 

Thank you for letting us know the issue, you've probably saved others a lot of grief.  You'd think they'd pass a "known issue" to those who interface with the customers so you wouldn't have to jump through hoops to get a problem identified. I'm also surprised you're the only one who's inquired via the forum.  Maybe others with the problem were informed it was a known issue?