Rather than building everything from source and run the risk of various breaks, running into a challenging dependency situation, or experiencing other hard aches, there's pre-compiled packages targeting Ubuntu that can be used for major kernel/Mesa upgrades outside of Canonical's small stable release updates. With pushing out new Ubuntu Linux releases every six months, Ubuntu doesn't issue major package updates after the release's debut, which is easier for stability and support, but not as nice when open-source graphics driver improvements are quick to materialize upstream and still working to compete with the binary GPU drivers and Windows graphics drivers.
First of all, I do not recommending using the 01.org Intel Linux Graphics Driver Installer releases or some of the other common approaches out there... The steps I am about to recommend I have found to be better for either being maintained longer on Ubuntu Linux releases, more quick to push down updates, or other benefits.
Upgrading the Ubuntu Linux kernel isn't as hard as it sounds, as long as you aren't relying on any binary drivers, don't need to use some out-of-tree kernel driver, or need any other special kernel configuration for your system. The easiest approach to upgrading your Ubuntu kernel is by using the Ubuntu Mainline Kernel PPA. From there you can fetch all of the latest upstream/vanilla Linux kernel releases... Including the very latest "daily" development snapshots of the Linux kernel. I commonly use this repository myself when wanting to grab the latest upstream Linux kernel and don't want/need to bother with a custom kernel configuration or any yet-to-be-merged patches.
With this mainline kernel archive, simply navigate to the relevant version/directory, and then download the relevant linux-image-* file for your 32-bit Intel or 64-bit Intel architecture. You only need the linux-headers support if you'll be compiling any kernel modules against that particular kernel, where you'd need the amd64/i386 headers to match your architecture along with the all version.
Recently the mainline archive began pushing out a lowlatency version of the kernel but most users will be fine with fetching the generic image. If I wanted to get the Linux 3.14 kernel right now for some Ubuntu Linux testing on 14.04 or any other recent Ubuntu release, I would go to v3.14-trusty/linux-image-3.14.0-031400-generic_3.14.0-031400.201403310035_amd64.deb. When that file is downloaded you can either double click on it to install it or use dpkg -i ~/Downloads/or/path/to/it/linux-image-3.14.0-031400-generic_3.14.0-031400.201403310035_amd64.deb. It should be that easy and when installing the kernel it will automatically update your GRUB boot-loader configuration.
Meanwhile, for the user-space graphics driver components, my recommended approach is to use either the Oibaf PPA or the Xorg-edgers personal package archive. Both of these repositories package the latest Mesa, libdrm, and X.Org graphics driver components for recent Ubuntu Linux releases. Upgrading these user-space driver components is as easy as running either sudo apt-add-repository ppa:xorg-edgers/ppa or sudo apt-add-repository ppa:oibaf/graphics-drivers for linking your system to either archive. That's then followed by running sudo apt-get update to update the package meta-data and lastly sudo apt-get dist-upgrade for upgrading all of your operating system's packages. After that, reboot, and if all goes well you'll be running the very latest Linux graphics drivers.
These steps are recommended for Linux enthusiasts and are fairly easy to carry out. I rarely run into issues running the latest Linux graphics drivers on Linux, but your mileage may vary based upon the hardware and experience. If you run into any problems or have questions, stop by the Phoronix Forums. Hopefully these steps were useful for those curious about upgrading the kernel or Mesa/X.Org on Ubuntu.