This project doesn't need a ton of parts, and there's some flexibility based on whether you want to use a headset or speaker&mic.

Start off with the basics:

Then you can either go with speaker + mic for 'hold it up and talk' style

You'll also need a microUSB cable, soldering iron and solder, and possibly some wire or other basic electronic components!

Learn all about it!

Before you put together the Arduin-o-Phone you'll want to read up a bunch and be familiar with the:

Arduino and Arduino IDE
2.8" TFT Shield - make sure you get this working nicely first
FONA 800 Shield - solder this one up with the stacking headers so you can stack the TFT shield on top! Then go through the example and make sure you have it working first

Wire & Test FONA shield

OK now you're ready to put it together. Most of the soldering happens on the FONA shield. Don't forget to solder it with stacking headers!

Attach the Mini Metal Speaker to the SPKR pins by soldering the + wire to the + pad, and same with the - wire
Attach the Wired Electret Microphone to the MIC pins by soldering the red wire to + and black wire to -
Solder the Vibrating Motor Disc to the Buzzer pins by soldering the red wire to the + pad, and the black wire to -
Plug the Lipoly battery into the JST connector, and switch the slide switch to CHRG
Insert the SIM card
Carefully plug the GSM antenna into the uFL connector, it should snap on nicely
Plug the shield onto the Arduino and connect the Arduino to your computer