Installing Gnome Shell on Linux Mint likely is not a good idea - see the comments also. Also this tutorial is deprecated; it's a year old at the time of writing.  But if you want to try, you could try this procedure. However, it likely won't work well.

I've thought about deleting the tutorial, but I'll keep it up there, as people wil likely want to try to install GNOME on their linux mint alongside xfce/cinnamon after all. This tutorial can stand as a testament that it probably won't work, at least not well. It's better to try another distro - you could also do that in virtualbox, or install a second linux on another partition, or even just run a live USB/DVD and try it out. But who cares? If you want to try it and it doesn't work it's easily uninstalled (you may want to keep track of all the components you install in a document or text file to help with a possible uninstall process). However - don't expect it to work well. 

If something changes and it does become easy to install GNOME - do let people know in the comments :) Or if you have a better way to actually get it work - post here and/or make a new tutorial. I'd happily remove mine ;)

More notes before you start (or to keep you from starting ;P):

Alternatives to this long-winded process: If you do want to try another desktop environment on top of your linux mint, cinnamon, MATE, Xfce and KDE should all work well in addition to whatever you have installed (which is either cinnamon, MATE, Xfce or KDE already). Also, if you want to try a gnome-based distro, you could try installing Gnome on top of Ubuntu, installing Ubuntu Gnome, or trying Fedora. I am sure there are even more - but I am not really a Gnome fan actually, and don't know...

Complications: A year ago, I didn't managed to install extensions other than the ones through terminal below. I'm guessing it is possible but extra complicated than in the original gnome version.

Personally, I've removed gnome again. I liked it, but it just isn't as compatible as I'd want it to be... By the way, when I installed gnome I installed it alongside cinnamon; I did not remove cinnamon.

Installing:

Installing the gnome desktop environment on linux mint can be problematic. On the forum I've read that a lot of people experience an issue where they try to login through gnome but the computer only returns a black screen with a moving cursor in the middle.

However, I found a way to make it work - or rather terminal showed me the way to make it work.

So here's my tutorial for installing gnome on linux mint 17:

Step 1: Installing through Terminal

Type the following in terminal:

$sudo apt-get install gnome-shell gnome-shell-extensions gnome-tweak-tool

When you do this, you will see that Terminal suggests a few "recommended packages". These are gnome-contacts, gnome-control-center, and gdm.

I found gdm was indispensable to make Linux Mint gnome work. This is a different login manager which will replace mdm (default mint login manager). When installing gdm, it will delete mdm. (At least, it did that for me). Gdm will still allow loading the original Cinnamon desktop environment. So no worries there.

So install all of them:

$sudo apt-get install gnome-contacts gnome-control-center gdm

There are also some "suggested" packages Terminal suggests. One of these is gnome-user-guide. I installed that one, as I thought it would probably be useful. I don't know about the other ones.
 (I'm a kind of newbie myself to all of this. Just been using linux for around 2+ months now).

$ sudo apt-get install user-guide

Step 2: Launching Gnome

Restart your computer. You will now see a Ubuntu-like start up screen.

Select gnome under the gear on the screen. This is under the login options.

Start it up.

Step 3: Tweaking Gnome

Open the Gnome TweakTool.

Click on Applications and click Search.

Then in the TweakTool, one of the first things you need to change is the settings under Appearance. Change Icons to Gnome, and other settings to defaults (Adwaita). (Or select something else. Defaults could be the safest though.) Keeping them Mint-X (most likely) causes extra vulnerability and instability. I noticed my Gnome system crashed a lot, especially with Libre Office, and changing this setting and window settings helped that a lot.

The other setting you will want to change is the window setting. You will notice that Gnome only includes 'x' on the windows by default. There are no buttons to minimize it. You can enable these on 'Windows' under the Tweak Tool. Under the "Title Bar Buttons" subsection, select Minimize and Maximize.

Fiddle with the other settings as much as you like :)

Also, you may want to install other themes. For this, the following YouTube video might be useful. http://www.youtube.com/watch?v=kNRVoQawlnk

Uninstalling Gnome:

$ sudo apt-get install mdm.

This will reinstall mdm (linux login manager) and remove gdm.

And "sudo apt-get remove" all other installed packages.