I would recommend Debian.
http://www.debian.org/
It is stable and reliable, which is what you want in a server.
Also, there is an enormous amount of packages available in the repositories (approx 30 thousand).
Also, the package management tools are, in my opinion, terrific.
Keeping your system up to date with latest security patches is really easy.
1) Type 'su' and press enter. You are asking it to let you become su - the SuperUser - also known as root. Only root can install or update software (this is a security measure to prevent others from messing up your system.)
2) It will ask you for the root password so type in the password and press enter
3) Type in 'apt-get update' and press enter.
This will check to see if any updates are available
4) Type in 'apt-get upgrade' and press enter
This will install any new updates
That is all you need to do to keep all of your software up to date.
There are 2 main ways to serve files on a local network - nfs and samba.
nfs is the traditional unix way:
pros
- it is very easy to set up
- it is extremely fast
cons
- has traditionally been very insecure (although if you are behind a router/firewall this may not matter too much - also, I gather that more recent versions of nfs are designed to be more secure than before)
- is not designed to work with Windows
nfs tutorials:
http://www.debianhelp.co.uk/nfs.htm
http://wiki.debian.org/NFSServerSetup
http://www.debianadmin.com/network-file-system-nfs-server-and-client-configuration-in-debian.html
Samba
pros
- designed to work well with Windows and allow sharing of files between Linux and Windows machines
- more secure than nfs (passwords required)
cons
- slower than nfs
- can be a bit more complicated to setup if you are not used to it
Really good explanation of how to use samba on Debian (parts 1 & 2):
http://archive.org/details/lrp025
http://archive.org/details/lrp026
More help:
http://www.debianhelp.co.uk/samba.htm
http://www.debian-administration.org/articles/340