I'm not familiar with Panasonics, but metering modes usually consist of:

• Evaluative/matrix metering (it takes readings from x spots all over the frame and averages them)
• Center-weighted average (like the first one, except that reading from the center influences the result more than the rest of the frame)
• Spot metering (takes reading only from small area of the frame, usually center, but it can also be bound to focus points in some cameras)

Evaluative metering is universally applicable for most situations on digital camera, where you can easily take a picture, check it (maybe even together with histogram) and apply exposure compensation.

You'll only need spot metering in very tricky lightning situations, where the subject and background are so many stops apart, that you need to apply more exposure compensation than supported by camera (usually +-2 stops) or don't even fit in your camera's dynamic range (ie when shooting moon).

When it comes to center-weighted average, I can't find realistic use case for it, as it is somewhere in between evaluative (which you simply have to trust) and spot metering (where you need to think about exposure), but you never know how smart the algorithm actually is and when it will interfere with your intentions.

When you want to delve into exposure and metering, your first step is to switch to spot metering and take the reading from different parts of the frame, think about how many stops is the difference and what you want to do about it.