First you need to decide what metering method best suits the kind of images you wish to make, bearing in mind the post-exposure methods you may be using.
It may help to think of what the Zone System does at its most basic level: it is intended to manipulate the dynamic range of the film to match the brightness range of the subject, while taking into account the photographer's intentions for the look of the finished image. The other simple way to describe the Zone System is the concise explanation of 'expose for the shadows, develop for the highlights'.
Think about what manipulations can be made during the development of colour film. Reversal (slide) film has a fairly narrow dynamic range when processed normally. Changing the length of time the film spends in the first developer affects both the dynamic range the film can capture and the density range of the image on the film. Within a small range this can be achieved with little trade-off in terms of increased graininess and decreased colour fidelity. Some films are better than others in this respect. You can pull most reversal film about half to one stop to increase dynamic range (lower the contrast of the film), and push up to about one stop to lower the dynamic range (and hence increase contrast and 'snap' - half a stop of push for this reason is not uncommon). If graininess is not an issue you can easily push further. If you are shooting reversal film it is best to get things right in camera (exposure, colour of light) and manipulate development if necessary. Most people will meter for the highlights, or for midtones in the well-lit parts of the scene (effectively the same result most of the time).
Modern negative (print) film has a very wide dynamic range when processed normally. If you expose for the shadows there will be very few scenes that result in blown-out highlights on the film. In many cases it may be better to place midtones in the preferred part of the film's response curve. You can use digital post processing to take the place of manipulation in development. You can also correct colour temperature issues more easily than you can with reversal film, especially if you have given the film enough exposure.
On to metering. Here's a problem. Meters are often 'inaccurate' when it comes to measuring strongly coloured objects, particularly at the ends of the visible spectrum. If all you want to do is to determine the brightness range of the scene you are in with a chance, if you use a reflected light meter in a sensible way - you should learn quite quickly how meter readings of various colours translate into results on film. You can make a very good estimate by measuring the range of incident light - Minor White covered this method in his Zone System Manual, and it has been further developed in the 'Beyond The Zone System' (BTZS).
I've missed a lot out, but I hope that there is something useful in all that.
Best,
Helen