I am new to Magento and I am about to build an e-commerce site for a client. Should I be using Magento's templating system to build the front end or should I consume it via the REST API? And is there another way to it? What are the pros and cons of each option?
If you plan to use Magento, then you should use it as it is.
I don't see any advantage in using the REST API just to retrieve data and show it as you want.
The REST API does not support everything Magento has to offer. It will also add a new layer between the frontend and the actual data. This may result in errors or it will slow down your website.
You should use the internal API and the templateing system. Magento is intended to be an e-commerce website.
But I'm very curious to what your other option is.