Yeah, generally you would put your connector of choice on everything, battery, car/esc, and charger. Be sure to work on only one battery wire at a time, don't short out the battery, don't let the wires touch.

And yes, the connectors go on the big wires. The small wires with the connector are the balance wires. Used to connect to a balancer or a balance charger. Allows the balancer to read the individual cell voltages, it is critical that no cell goes above 4.2 volts. Or below say 3 volts to be safe, certainly not below 2.7. 3.2 volts is even safer.

As far as number of cells, does it have a voltage rating, like 7.4 volts? Lipos are generally rated at a nominal voltage of 3.7 per cell. Your speculation of 2 cells from the visual description is probably right. Take a look at the small wire/balancing lead. Is it 3 wires? That would be for a 2 cell, positive, negative, and a wire in between the 2 so you can get the individual cell voltages. A 3 cell would have 4 wires and so on.

If you don't have a balancer of some sort, suggest using a voltage meter to keep tabs on the individual cells as they reach full charge state. Overcharging causes the fires. You are using a lipo charger right? Discharging too low causes the cells to die. If your speed control does not have a low voltage cutoff, be sure to stop immediately as soon as you sense any reduction in power. Don't even drive the thing back to you, go and get it, and unplug it.