Hi,
I just installed Slack 10.2 on my machine, with LILO as the boot loader. But when powered on the machine, it showed the GRUB prompt, with no option for the OS boot images. I thought LILO would have wiped off GRUB, but it seems to persist, and I can't seem to find a way out. GRUB came from my previous install of FC3. I don't want to use a boot disk everytime. Is there some way to weed out GRUB? Please advise.

I'd rather use grub...but if you want lilo, you have to run:
as root to (re)install it with the changes you might have made to it's config-file (/etc/lilo.conf)
The config-file needs to be right though for this to work - (for lilo being installed to the MBR that is...) - so that you can write lilo to the MBR and thus boot off the device (using lilo) you want to boot from.

Frankly speaking, even I'd use grub, but that's only possible when it shows me the boot options. I don't want to mount the kernel images manually everytime I boot. What do I do?

You want to use grub?
Do you have grub installed - I mean the binary, not the "leftover" in MBR from the previous FC3 install?
tells you.
If you do - do you have a separate /boot partition? If yes - mount it to /boot on your / filesystem.
Edit the configuration-file /boot/grub/menu.lst to match your current setup (name of kernel image, where is /boot/grub with all the stage files and the config-file in it, which is your / partition)
then run:
if /dev/hda is the disk you want to boot from
Any questions?
...post your disk-layout and grub-config (menu.lst) along with it...
also this could assist you: http://www.gentoo.org/doc/en/grub-error-guide.xml