How does the skill block work. Do all of the perks in the block skill tree work with a weapon also? Can I bash while only using a one handed or 2 handed weapon?

Left side of the tree is more shield oriented, right side has good perks for 2h and shield.
It won't block arrows until you get the perk.
I think you can bash with a one hander if you hold it with 2 hands. Not positive.

You can bash with a one-handed weapon or a bow, both giving experience toward Block.

You can block with a shield or two handed weapon. Shields are much more effective. Most blocking perks work for shields and two handed, but some work only for shields. The latter perks' descriptions specify that they are for shields.

Check out this table: http://elderscrolls.wikia.com/wiki/Block_%28Skyrim%29#Perks