Aquatroy is right, if you were a heavy smoker, the short battery life of the cig-a-like types will grow old on you quickly.
That said, to directly answer your compatibility question, it's all about the threads on your device. Eversmoke uses a KR-808 D1 thread. So, if you wanted to try pre-filled cartos from another company, just make sure they have the same KR-808 D1 threads.
As for the batteries themselves, I do not know. Blu cigs used a pretty standard 510 thread, BUT the battery's polarity is reversed, so you could not charge the Blu battery on another brand 510 charger and you could not use another brand 510 threaded battery on the Blu charger. I do not know about the Eversmoke's polarity, so best be on the safe side and just suggest that, if you get a different battery with the same threading, by a new charger with it too.
However, I must go back and suggest what Aquatroy said and say you should look for a better vaping starter kit. Discount Vaper has a couple of really good eGo starter kits for right around $50, and the eGo battery is a good starting battery as it supports two thread types: the popular eGo thread which most clearomizer delivery systems use, and also the 510 threads which a lot of carto manufacturers use. Good luck!