@op
You asked the right question, and this is the right place to ask it if you want the answer. liudr has a point, but where else would you go to get the answer to that question?
The question: picaxe or Arduino?
The answer: Either one will do the job. This is not a particularly difficult project hardware or software wise, but you will run into issues related to interfacing with your car. No, you won't need any kind of analog to digital board--both the Arduino and the picaxe40x2 have microcontrollers with built-in ADC pins. The more important issue will probably be which one is easier to learn for your first project? If you already know BASIC or C++, that might help. Both pixace and Arduino have very active forums and lots of tutorials and documentation. It's probably harder narrowing down all the information to something you can use than finding the information in the first place. Arduino probably has a more active user space. If you can put up with (or ignore) the sometimes gruff attitude you'll get here, we can help you do this project.
The question: Is there a kit you can buy?
The answer: Well, kinda, but not exactly. There are starter experimenter kits with all sorts of different hardware, and projects that use that hardware. There are some with LCD displays, and some with temperature sensors. I didn't see any with both. If you buy an experimenter kit, you are pretty much guaranteed that you can get the hardware that comes with that kit to work. The downside is you get lots of other hardware that you won't use, and it's more expensive.
This is a pretty good first project. There are basically three things you need to learn how to interface with the Arduino. The Temp sensor, the LCD screen, and the voltage sensor.
I recommend you get the Arduino, LCD screen, and temperature sensor. Also a breadboard, connector wires, and an assortment of resistors. If you get them from Adafruit, they have excellent tutorials that can get you up and running easily. First, learn how to make the LCD screen say what you want. Then, learn how to get the temperature from the temp sensor and put it on the LCD screen. Then, learn how to figure the voltage. If you have problems, we can help.