if you're a newbie, I suggest getting a binary kernel package (like rpm for Fedora) that you can install by running it / double-clicking / something like that. I don't know what package manager (if any) Mepis uses, but use it if you can.
If you really want or need to compile the new kernel from source, go to kernel.org and read the docs...it's really a simple thing, but you need to know a few things first and make sure you have the dependencies (like gcc). After getting and extracting the source, it's really a matter of few commands to get the new kernel.
And in Linux, you can compile several kernels and put them in your boot partition side by side and choose which one to boot from - if the newly compiled kernel doesn't work, just reboot and select the old.
So, first see if you can use somekind of package manager (how you usually install programs in Mepis?) to install the kernel. Only if you cannot use that, then go to
www.kernel.org and download the sources for a kernel you want (some couple dozen megabytes). With the kernel source code you get a README-file which tells you what you need to compile it and how (normally: make menuconfig - make all - make modules-install - copy files to your boot partition, or something like that...it's all in the README).
EDIT: just to remind - a kernel is also a file in your disk. So as long as you do not remove your old kernel or replace it, but only add the new kernel (copy it) with a different name (the version number makes the name different) to your boot partition, and edit grub's config file and add there the new kernel by copying the lines of the old and pasting them and editing a bit, you have 1) two kernels, the old and the new and 2) two entries in Grub for the two kernels so 3) you can select either one of the two to boot
