I leave the bridge and pull the wick. I've never completely pulled the wick, I leave the part that runs between the coils. Just pulling the obvious chunk of wick directly under the bridge has been good enough for me.
some quick tips... Most 306s come primed, Cisco brand do not. When you get a new one, pull the wick and then blow it out into a napkin as well as you can. If you can, certainly not necessary, run hot water through a new (primed) atomizer before using it, blow it out really well, then let it dry overnight. If you have a fan you can set the atty on, great... if not, it should air dry. Again, not entirely necessary... you can get most of the primer funk out by just blowing it out.
A completely clean and dry 306 will take 7 drops. When the flavor starts to change, add 2-3 drops. To change flavors, just blow the atty out into a napkin. After blowing it out, the atty will then take 5 drops... and 2-3, as usual, when the flavor starts to drop off....
There are mixed reviews on dry-burning, but I've been doing it for 5 months and I have a few atties that are still going strong after this amount of time! Search "my atty resurrection method" and follow it when the coils get black and funky. Usually takes about 7-10 days before the 306 needs to be cleaned. Works wonders on the 306s. Just be careful!