Assuming there's something on the print bed, I usually look at at what the result is to determine what to check on the printer next.
Before doing anything physical with the printer, check your settings. You should have a decent understanding of what changes what and how the different numbers affect the print, as well as what they "should" be. Review them and make sure there's nothing that looks out of whack.
Now for the specifics:
If it looks like the filament didn't extrude right, for example if it stopped extruding halfway, barely extruded at all, or there's gaps in the print, I would check the hotend. Reload the filament and push it through by hand to make sure it's not jammed. If the nozzle is fine and it still looks like you're having extrusion issues, I would check the following things:
- Wiring to the motor
- Excess tension on the filament before it goes into the extruder (bowden/guide tubes, etc.)
- The nozzle itself. Nozzles do wear out and they're relatively inexpensive so if I can't solve an extrusion issue with reloading the filament, the next thing I do is replace the nozzle.
For mechanical issues, it's a little more complicated. The very first thing I do is pluck the belts on my printers and make sure they're tight. It should play a low note somewhere around a G. If it's a thud, the belt is too loose. If it's a really high sound, the belt is probably too tight. Another thing I check is to make sure all linearly moving parts (such as the print bed, carriages for the nozzle, etc.) only move in the direction they are supposed to. Wiggle them back and forth and make sure nothing is loose on the printer.
If you don't find anything that would obviously indicate what the problem is, run the print again and watch it closely. If it's only messing up in a certain part of a print, make sure you watch what the printer is doing when it gets to those areas. Make note of the nozzle distance to the bed at startup, retraction and hopping during prints, and making sure no wires or anything are getting snagged. Listen to the printer too, if it's making any new noises such as clicking or grinding then you most likely have a problem on your hands.