Your oil burner will not light up and by checking the transformer you find out that it is not making the spark needed to light the burner. Changing this part on is not difficult. I will give you a few pointers on how this can be done easily.
To check you ignition transformer quickly, just loosen the screws that hold it down and flip the little tabs back. After pushing the reset button wait for the motor to start. Once the motor starts, pull up on the transformer a bit and listen for the spark to jump to the electrodes. This spark is normally very hot and will jump a gap of at least ¼”. If you have a lazy yellow/orange spark instead of a bright blue/white hot spark or if there is no spark at all then you need to replace the transformer.
To change the transformer turn off the power to the burner, and flip the transformer all the way open. There are two screws that hold the hinge at the back of the transformer. Remove the protectorelay from the junction box and unhook the wires going to the transformer. Remember where these wires go so that you can reconnect them to the same spot.
Fasten the new transformer in place using the two screws and connect the wires to the proper places. Turn the power back on and try to fire the burner. The reset button on the protectorelay may need to be pushed again. If the burner does not light then you may have other problems also. You will need to continue to troubleshoot the problem or call professional help.
It is not uncommon for oil burner problems to come in bunches and for more than one part to fail at once. Do not be discouraged if there are multiple problems. This is very normal with oil burners. Have patience and methodically check each component and replace as necessary.
Oil burner repair can be very challenging but there are many things that you can do if you have some mechanical knowledge. Have fun and good luck!