This is all in books online. Hopefully you can try in development, or on your desktop first. There are quite a few things to consider, so you should give it a thorough read.
Yes, you can mirror without taking it down. You restore a backup to the mirror and use NORECOVERY option. You then setup the mirroring and transactions start to move from primary to mirror. You'll need to set up any other things needed on that remote server - logins, jobs, etc.
Transfer logins using this method, so you get matching SIDs between servers.