How do you travel around world, and is there some quick method, like the flying chicken?
There are several ways to travel in CoS. Walking and horseback are the most obvious, but there are many types of mounts, including various ground mounts, air and water mounts. However, a note about horses...
In CoS horses are not something you must wait to obtain. You can always get an 'issued' horse from the Citadel Garrison. These aren't the best horses (by far), but you can get mounted for free, right away. As soon as you can afford a better mount, get one. Better horses do cost more, but you can get a basic horse right away.
In addition, mounts, are more 'real' in CoS. You must mount and dismount, and mounts do not disappear. They must be picketed or tied to a hitching post. They will, however, come when you whistle. Finally, mounts will not go where they are not supposed to go. Depending on the quality of the mount, they may balk at jumping down too large an incline, or crossing flowing water, etc. However, no mount will go underground or in structures. OK, enough about mounts for the moment, and back to the meat of your question.
CoS also offers Permanent and Personal portals. A Permanent portal cannot move, but will take you to any other Permanent portal that you have previously discovered or to your Guild hall, or to a personal portal of a group member.
A personal portal can be opened (for a small charge) almost anywhere. These portals will always return you to the Citadel, and then back to the location where you opened the Personal portal.
Beware portal blocks, for they can trap you in a zone where you cannot open a Personal portal to escape.
As for flying and water mounts, and other forms of transportation... there will be some of these features in the game, but we're keeping those details a secret for the time being.