It's possible that grub just didn't get installed properly during the install. (Happens sometimes.
) Try following to re-install grub:
* Boot from live Linux Lite DVD/USB
* Open a terminal (Menu -> Accessories -> Terminal)
* Mount the installed system's root partition to the live enviroment
sudo mount /dev/sda1 /mnt
* Install grub to the MBR (master boot record) of the drive
sudo grub-install --boot-directory=/mnt/boot/ /dev/sda
* Unmount the root partition
sudo umount /mnt
* Close terminal, shutdown computer, remove DVD/USB
* Boot computer again and see if everything now works.