To move single cards:
On the card's page - you'll see the collection your profile is currently in above the area where you can put in the checkmark to put it in your collection (the collection section of the card page). Remove it from that collection, switch to the collection you want it in by clicking on the collection name and selecting the correct one from the list, and add it to the collection there.
To move all cards in a set:
Go to the Collection Summary page for the set. In the "Choose an action" drop down, you'll see all your collections listed there at the end of the list - the list will have "Move all items to [collection name]" with as many collections as you have.
To move a single team in a set:
Go to the Collection Summary page for the set. Select the team filter. In the "Choose an action" drop down, you'll see all your collections listed there at the end of the list - the list will have "Move all items to [collection name]" with as many collections as you have. Choosing to move all items only moves the ones displaying (I just tested this).
-- Dan --
Note: Please see my profile for more info regarding trading (section updated 3/4/2024). I have added a large portion of my inventory to the site, and currently have trading turned on (details are in my profile).