Garage door button with delay feature
I figure out a solution - I just needed to use a relay.

I didn't know what a relay was, and had a hard time figuring it out.  I've included some notes for anyone who finds this post in the future.  


I can't determine if it took reading 3-4 articles on relays for it to sink in, or if one article was the best.  Anyway, for me this one helped the most: https://www.instructables.com/id/How-Electronic-Switches-Work-For-Noobs-Relays-and/
My relay looked different from most of the tutorials.  I figured it out by connecting a couple alligator clips to a 9v battery and listening for the click of the magnetic switch.  Then I used my volt meter to determine which leads were working with the relay.


Relay Setup:


I connected my ~5 to one lead on the relay, and 5v to the other
The Common (middle) lead connects to one terminal on the garage door opener
The N/C (Normally Closed) lead connects to the other terminal on the garage door opener


Button Setup:


The button setup was just like all the other button tutorials
When the button is pressed (LOW), I start and stop the relay to trigger a "button press" for the garage door opener (it starts)
I then run a delay for the set amount of time I want
All button presses are ignored until after the delay has finished


Now I can move on to the actual project.  Thanks!