Which is a better starting point between the two? I have not done either, and am not sure how much gear I need for heroics. There are several good guides for getting Kara ready, but I've never seen anything on getting heroic ready. Are the two equivalent in difficulty or gear requirements?
Hard to say without knowing your gear but I would recommend going for the easier heroics first – Slave Pens, Underbog, Mechanar (get Sun Eater), maybe Botanica. Get the pvp shield. Get some badges together, buy some gear, head off to Kara.
At least that's what I did. But after patch everything became so easy so maybe just head straight for Kara.
i'd do the base heroics first - SP, UB, Mech. Learn the 5 mans and tanking the different types of mobs and the different classes in your groups. Do kara once or twice to get some gear then spend whatever time left you have in ZA.
I've done mech 11 times and still no sun eater so get your sso rep and use innuros blade for the time being
Pretty much always do whatever heroic daily there is. Kara is a saved instance so you can only do it so much.