If you've already got some media from an established tank, then thats generally the best way to go about starting your cycle in a new tank. It'll cut down on the cycling time some for sure, however its not an instant or perfect fix, especially when you're going from a much smaller tank to a larger one, so you will need to still watch your levels closely and be ready to preform water changes accordingly for the first few weeks/couple of months until things stay stable for at least a week or so. Then you can start going down to regular weekly changes. ;)
Like I mentioned above, using old tank water doesn't do anything. There isn't enough BB in there to properly seed a new tank, and whatever ammonia is in that water will just be watered down. You need some kind of concentrated source of ammonia.
Transferring substrate/decor/live plants and all that will certainly help a little, but the majority of your BB colonize in your filter media so adding in that is what is going to get you the best, most efficient, and quickest results when we're talking about seeding. And its going to take a lot longer then a day for the BB to grow and colonize on the new and larger surface of your new filter/tank, so you'll probably want to keep it in there until your tanks cycle has stabilized for at least a couple of weeks.
Now, what I would do is instead of using the filter cartridge with the carbon in it(which is actually useless unless you're removing medication from your tank, and I've read can actually remove some beneficial minerals from your water during the short period its actually active) in your new filer is pick up some filter foam/sponge. Take a piece(or two, depending on how large they are) and wrap it around the filter media that you will be using to seed, then put it in your filter. If there is any extra space, stuff it with more sponge/foam. This will give you some more surface area for your BBs to cling to and colonize, and unlike the cartridges where the floss breaks down eventually and you have to replace them(which essentially just throws away most of your BB anyway and could easily crash your cycle)you shouldn't ever need to replace the sponge/foam(or at least it would be a VERY long time before you might need to), which saves your cycle AND saves you some cash in the long run. Just take it out every water change/every other regular water change(AFTER your cycle has been established), swish it around in some old tank water, then stick it back in the filter. :)
Haha, Corys are definitely little characters, especially when kept in their little shoals! So cute. I would suggest a sand/very fine gravel substrate for them as they like to dig and root around and sand is a little softer on their barbs. ;)