Hmm..Ubuntu, unless configured otherwise, should have it's root account locked. Instead of having an active root account it uses
sudo (which is really a sensible way) and grants the first user created during the setup full sudo permissions by default (new users created after that need to be given sudo permissions separately). Therefore if you can't log in at all with your user account you need to either use a live-cd or boot into single-user mode to fix this problem (not sure why your home directory does not exist).
To boot into a single user mode: when you boot, GRUB should launch either showing you a list of installed kernels or a message asking you to press ESC to see the meny. Press ESC if needed, then
e to edit the boot lines (you should not be asked for password, or if you are, you should know that password -- bootloader password). Edit the kernel line, and at the end of the last word in the line, put one space and type
single. Press ENTER and
b which should now make your Ubuntu boot into a single user mode, into a maintenance runlevel, and you should be logged in as root without any passwords (this is why you should set a bootloader password, to prevent this from happening if somebody gets to your machine). Now that you're in, and know the username who's home directory is missing (let's call it
username in this example), first create that missing home directory:
Code:
mkdir /home/username
then set the owner to be
username (it's root by default if you create it as root user):
Code:
chown username /home/username
After this the directory /home/
username should be ok (that's exactly the same directory you got warned of). Try to reboot your machine the usual way and log in as the user you did in the first place and see if that helps (if not, post a message here and describe what happened).
It could be that somebody had accidentally removed the home directory. You should still be able to log in normally, but you couldn't probably write anything anywhere since the regular user should not have permissions to write directly in root (/). That's why home directories are used; users have full permissions to their home directories, and not elsewhere; it's more secure that way. By default when a new user is created, either by using the graphical tools Ubuntu's desktop provides, or by the command
useradd, a directory of the same name as the username is created under /home and set as this user's home directory, unless specified otherwise.
Hope it helps. Ubuntu is a great choice to start using Linux

if you need more software to it, and have an internet connection, use
apt-get or one of it's more user-friendly graphical front-ends, found in the system's application menu,
Synaptic for example.