It is possible to install your own kernel from the mainline, but not recommended.
Unless you have some specific piece of hardware that doesn't work in 3.8 but is supported in 3.11, the changes are very unlikely to make a difference to you.
You can read a (shortened but not short) list of changes at eg, KernelNewbies. The Linux kernel as a whole supports a large number of different computer architectures and peripherals and hence most changes probably don't affect the hardware you are using. Changes to the core might make minor differences in eg, file system performance but in most cases you won't notice the difference.
In addition, installing your own means you don't get the advantage of the package manager's scripts which streamline installing a new kernel - for instance, you'll have to manually update grub to boot the new kernel and manually install any proprietary drivers (nvidia, broadcom, etc) you were using.
If security issues are fixed in newer kernel versions, they are patched in the currently supported Ubuntu versions and you get these fixes by running a normal Software Update.
Ubuntu 13.10 Saucy Salamander will be released in the next few weeks with a 3.11 kernel.