Axemen can also perform barb-busting duties although not as well as chariots in my opinion.
Chariots can also perform good double duty as fogbusters since they have access to the sentry (+1 sight range) promotion.
When you build your second city, if you are not creative, make sure you are claiming a military resource in your inner ring. Long term potential of a site means nothing if you don't get to the long term. The earlier you get the resource hooked up and start pumping real units, the less chance you have of getting pillaged or set back in any way.
No barbarians is a really lame option because it will be infinitely harder to get well trained troops in the early going.