Welcome to the hobby.

Basic battery stuff:
5000 mAh: this is equivalent to how big the tank of fuel is. More mAh (milliamp hours) = the longer the battery will run for.
3S = number of cells. More cells = more voltage = more power.
11.1v = nominal voltage. It can be charged to 4.2v/cell if it is a normal LiPo, so 3 cells x 4.2v = 12.6v.
C rating = basically marketing garbage. Higher C ratings mean the battery is capable of delivering more amps, but the ratings are not consistent enough and often inflated. In general, it is used as a measure of quality (more C = more better).

LiPo safety basics:

1. Never let it get below 3.0v / cell (3S = 9.0V), and store it at 3.8v / cell (3S = 11.4v).
2. Always balance charge! This means you keep each cell near the same voltage as the others while charging. Almost every charger balance charges by default, so just don't go cheap on a charger and don't bypass the balance charge setting. If you don't keep them balanced, one cell may go to low or too high and then the battery becomes a fire hazard.
3. Charge and store your batteries in a LiPo bag (fiberglass) away from combustibles at a minimum, get a Batt-Safe or set up a concrete block/ammo can for charging and storage.
4. Do not charge them unattended.
5. Do not store them in the vehicle (hard to tell when they swell or puff and become dangerous).
6. Do not leave them plugged in when not in use (slow drain can lead to instability and fire).

Now on to your question:
Which ESC you need will depend on the size of the vehicle and how it is used. A lightweight 1/10th scale 2WD buggy can use a much different ESC than a monster truck. The motor size and weight of the vehicle will determine how many amps the motor will pull, and the ESC needs to be able to handle at least that amperage.