router question from dummy

Have high speed internet. Have a 3gbsdram. And I have a router that is a g.
Would I benefit as far as computer speed on the net with a router that is an n router. Also even though the computer says 802.11 a/g/n wireless I still would have to buy the the adapter that goes into the notebook?
I know nothing about routers and the fact we have a g router it doesn't have an adapter needed for the computer to work. Was wondering if that is the same case with a wireless n router? thanks