Im not sure if this post should go here or in the habitat's section, so I appologise if it is in the wrong place.
After reading the very helpful post on tank cycling, what I was wondering is, would it be ok to move an ornament or piece of bogwood from my coldwater goldfish tank into my new tank that I will be setting up for my very first betta to start to bacteria population/cycling off?
Unfortunately, only a small percentage of bacteria is present outside of the filter. This is why you can do large water changes without crashing your cycle. The bacteria we cultivate during tank cycling requires oxygen to survive, and this is why it sets up shop in your filter.
What you need is seeded filter media, which (hopefully) will contain enough beneficial bacteria to speed up the cycling process. If your goldfish tank is cycled, I would take out some of your filter media from there and put it into your new filter. This process isn't always exact, and so you will need to monitor your water parameters until a colony has established itself.
You will also need to provide an immediate source of ammonia (either a fish, or a bottle of pure ammonia) to prevent the bacteria from starving.