Thermal IR cameras are fantastic for locating likely trouble spots. You can quickly scan a large area with them, see hot spots, and usually get enough detail that you more or less know what you're looking at. Other methods take far more time and/or are not capable of measuring small components (most IR spot thermometers have a spot that is enormous, compared to the size of small electronic components).
For precise measurements, though, contact probes are more trustworthy, especially if you're going to be measuring on metal, provided you can hold or mount the probe to make good contact with the component. Less so on plastic, where it might take a while before the plastic heats up the probe enough. Low thermal mass probes are preferred for that reason.
Emissivity correction for extra precision... Usually, I'd say, don't bother, just add patches of high emissivity black tape to any shiny surfaces (for really hot things, heat-resistant matte black paint). There are two reasons for this.
- Emissivity tables are available, but they're rough guidelines only. The emissivity will not only depend on what kind of material it is, but also its surface state (oxidation) and roughness. This isn't always easy to estimate.
- Emissivity is coupled with reflectance. The lower the emissivity, the higher the reflectance, so the part of the heat the camera sees, that is not emitted, is reflected heat from the environment. This means emissivity correction requires two parameters: the emissivity of the material (which you don't know exactly), and the temperature of the reflected environment. That's fine if you are in an environment with a perfectly homogeneous temperature everywhere. Not so good when the environment has all kinds of hot spots nearby, e.g. light fixtures, other heat sinks and equipment, humans,...