Along the way to the very expensive Horseback Riding is Animal Husbandry for horses and Chariots. The latter may be sufficient for your initial barb-thumping needs.
Granted, Chariots are weaker than Axes (4 instead of 5), but they're cheap. Build several of them--at least two per city--and as they fight and win, give them Combat I and then Shock for a bonus against Melee units (like Axes). If they die, attack the weakened barb immediately with another one, and then replace the lost unit ASAP.
Remember to wait until the barbs are on open ground (not on hills or in jungle/forests) and on the Chariots' side of a river before attacking, to make sure they're not getting any defensive bonuses and thus increasing the Chariot's chance of success.
I've been able to defend myself from Barbs very ably with Chariots, so much so that HBR is usually unnecessary.
As for Archers--why leave all of them in your cities? Give a few of them Guerilla and/or Woodsman promotions. Fortify them in appropriate spots (Guerillas on hills, Woodsmen in forests, duh) on the edge of your territory, wherever the barbs are coming from. The barbs will likely attack them and not your resources, and the barbs will almost always die, since the Archers have a bonus from the promotion, the terrain, and fortification. They'll also push back the fog and reduce the number of barbs getting generated.