At this point, you could try to disconnect the screen assembly (disconnect the battery first) and try connecting it to an iTunes enabled computer. If the phone is not recognized, then you probably have a logic board issue. This is much harder to troubleshoot and considering you don't have a multimeter yet, it is probably beyond your skill level at this point in time.
If you still want to proceed, then get your multimeter and then find the schematic online (sorry we can't share them here) and find PP_BATT_VCC, PP_VCC_MAIN and PP5V0_USB. I would start by checking to see if those power rails are shorted to ground (device has battery disconnected). If one of these rails is shorted to ground, then you will need to identify what is causing the short. It could be a bad decoupling capacitor, conductive debris or defective IC that is directly supplied by those rails.
If they are not shorted, then you can connect, preferably, a known-good battery to see what voltage you measure. If you are measuring the proper voltage, then you move onto the PMIC and check the voltage rails it generates. But that is pretty advanced and you should report back before going too far.