I wrote something similar to what Rod did but didnt post it. I agree that you have to at least understand the basic operation of the machine. You also have to understand how the components used on the machine works.
Assuming you have a basic idea of how the machine works the next step is to determine at what point the problem occurs. In many cases you dont need to hook a PC up to see the ladder, those little lights will tell you ALOT....ie you push START button but it doesnt start, you look at PLC and input for start doesnt light when pushed so you then check for voltage at the PB and/or continuity thru the PB, the results determine the next step. Thats why its always nice to have a drawing or at least a P&ID available, drawing is better.
There may be times when it seems that all I/O are working properly but the machine doesnt perform properly. THEN you connect a PC to the PLC and go online>then find the section pertaining to WHERE the problem occurs...the next step depends on what you do or dont find.
Troubleshooting is like art or music etc, its a skill that is developed over time. Like art, music etc some people seem to have a natural talent for it but it still takes time to truly develop.
One way to make troubleshooting faster is take the time to learn the machines. When not busy spend time at the machine(s), talk to the operators, help them run it and learn how to run it.
Another option that can help is attaching an HMI and creating as many "alarm" messages as you can. Its so neat when the screen tells you CF2 no voltage...ie control fuse 2 is blown.