So I bought a PIC Programmer (it's just an amazon's "user rated" PICKIT3 clone) and tried to read it with MPLAB X IDE v3.65 (and MPLAB IPE).
I've turned on the "Power target circuit from PICkit3" option and connected the PIC more or less as in this schema (except for the led and buzzer that are absent and a 2200uF electrolytic capacitor between pin 2 and pin 3):

the PIC programmer seems to be working or at least is well recognized from MPLAB but the log says:
Connecting to MPLAB PICkit 3...
Currently loaded firmware on PICkit 3
Firmware Suite Version.....01.49.09
Firmware type..............Midrange
Programmer to target power is enabled - VDD = 5,000000 volts.
Target Device ID (0x0) is an Invalid Device ID. Please check your connections to the Target Device.
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Giving that this is my first time with a PIC programmer, do anyone have an idea of what could I be doing wrong ?