Whether you need to install anything for the Wi-Fi depends on the brand and model of the Wi-Fi chip. If that information is not readily available on the vendor's website, we can help you figure that out once you have the laptop and you have booted into Linux Mint (okay, my preference
as Linux Mint has a couple of extra tools installed to help with this, as compared to Ubuntu).
Hopefully you don't need to install anything and it will just work. It will be handy though to have a network cable available so you can use your laptop with wired Internet if the Wi-Fi needs some work.
You can install Linux Mint without an Internet connection, but for best results it is recommend to have one during installation (even if just temporarily with wired Internet for the installation).
There are no special steps for installing an operating system on a computer that doesn't yet have an operating system. In fact it makes the installation easier, as there is no risk of accidentally overwriting the other operating system or such. If your laptop is certified for use with Windows 8, you may need to go into the BIOS and disable secure boot. I don't think that is the case though.
With a 2.4 Ghz Celeron processor I would venture that you won't have the best performance with a more demanding desktop environment like Ubuntu's Unity. I'd recommend Linux Mint 64-bit with MATE instead. It depends a bit also on what graphics card you have?
You can download Linux Mint editions from here: http://www.linuxmint.com/oldreleases.php
. You can either burn a downloaded ISO onto a DVD, or you can put it on a USB stick with these steps: http://community.linuxmint.com/tutorial/view/744
. Linux Mint 13 is the long-term support release, which you can use till April 2017. Linux Mint 14 is the latest release, with newer software and drivers, which you can use till April 2014.
As for putting home on a separate partition, yes you can do that but it will complicate the installation a bit. I'd be happy to add the manual partitioning steps here if you want, but Linux Mint's recommended way of installing a newer release (http://community.linuxmint.com/tutorial/view/2
) will have you overwrite that partition upon installing the newer release.