The country code for Mexico is +52. Every Mexican telephone number has an area code (the code of the city where the phone was bought).
In total, a Mexican cell phone number has ten digits. Add 044 if you are calling a cell phone from a landline and if you are dialing from the same area as the area code. If you are dialing from outside the area code, dial 045 instead. If you are calling from a cell phone to another cell phone, dial the ten digits. To call a Mexican cell phone from another country, dial +52 followed by 1, and then the ten digits. If you are traveling for a longer time, and if you expect to make various phone calls within Mexico, it might be worth buying a cell phone in Mexico. Roaming with your cell phone in Mexico may be possible, but this can be pretty expensive. Be aware that once you are away from your area code, your phone calls will be more expensive.
To call a landline number in the town you are in, simply dial the local number (no area code needed). For a landline number in another town, dial the long-distance prefix (01) followed by the area code and then the number.
For international phone calls, a couple of options exist. Some internet cafés and hotels offer this service. Alternatively, buy a long-distance discount card. The cheapest option is probably to use skype.