How can I turn an iPod touch into a functional mobile telephone? Considering that the iPhone is three times more expensive than the iPod touch and that the iPod touch is much thinner than the iPhone, it would be quite neat to use an iPod touch as to send and receive phone calls.

What would need to be done to enable an iPod Touch to send and receive phone calls (either Wi-Fi only or better interfacing to cellular data through an accessory)?
There are three steps that would need to be completed to enable an iPod Touch to act as a mobile telephone.

First, it would need to make phone calls. The answers to this question document how to place phone calls using an iPod Touch and VOIP. I personally have used Talkatone, and find that it does work, but has some substantial lags, particularly at the beginning of the conversation.

Second, it would need to receive phone calls. (Note that this is a key difference between using an iPod Touch as a mobile telephone, as asked in your title, and a calling device, as asked in the body of the first version of your question!) This is a much more difficult prospect, as presumably you would want the device to be able to receive calls at any time, and not just when a particular app is running. It would appear that the program Skype is available for the iPod Touch and could be used to make and receive phone calls.

Finally, unlike the iPhone, the iPod Touch can only communicate with the rest of the world when it is connected to a WiFi network. A mobile hotspot like the MiFi carried with you could give the iPod Touch steady access to a WiFi network.

All that said, you'll end up paying for Skype in and out on top of paying for the data plan for the MiFi. It might be simpler and not a whole lot more expensive to just go with an actual iPhone.

In other words, it can be done to make an iPod Touch more like an iPhone, but that doesn't mean you'd want to, besides as a proof of concept.