Newbie,
The Bakelite caps with the spring under them are the brush caps, do not oil or grease them. That would really mess up the motor.
The grease caps are on the bottom of the motor. Two large metal slotted screws with knurled edges. Lay the machine on it's side, remove the slotted metal caps and stick the end of a tube of SINGER lube in the hole and gently squeeze it in. These big holes lube the motor bearings and the drive gear at the same time. So don't go nuts with it, but don't be chinchy either.
If the wiring is good, check it thoroughly, then the motor is most likely just as filthy inside as the machine is. Lube it, then run it. It should clean up inside.
To clean the machine externally use oil and cotton balls or flannel cloth. Underneath same thing. Oil everything that moves other than the motor.
There is no need to take it to a service person unless you just can't work on them.
Joe