This is what I found when searching for your laptop.
Sony VAIO PCG-F801 - 12.1" - K6-2+ - Win98 SE - 64 MB RAM - 6 GB HDD
That 64mb of RAM might be your problem. Zenwalk requires 128mb of RAM as a minimum.
You might want to try AntiX or Puppy Linux. Their minimum requirements might fit the 64 mb of memory. Myself I would try AntiX, as I have installed it on an old Thinkpad i1413 which had low memory and a 6Gb hard disk. I forget how much memory it had, but the creator of AntiX has said that it will install with even 64mb, even though that would be pushing the minimum requirements.
Home page for AntiX is here: http://antix.mepis.org/index.php?title=Main_Page
Download mirror (One of many, but I'm posting the link so you know which version to download):
Look for this version - "antiX-M11-486.iso"
Since you're learning, you should try the version I suggested, although on the AntiX home page, it states that machines with K6 processors might have to use the new Core version, then build on that until you've reached the limits of memory. Hopefully, the M11-486 version will work out of the box for you.
The other problem that I see you might be having is installing it on a laptop with one type of video card and then moving it to a laptop with a different video card. While in normal circumstances this shouldn't be a problem, the low amount of memory might be hindering the video. Sometimes adding the vesa cheatcode to the kernel line at the grub menu might help with that problem.
It's difficult explaining to a beginner, but, when Linux is booting up, the grub screen, which contains the kernel that is being used, comes up for a few seconds. It's one of the first screens that you see during boot up. There will be options listed at the bottom, such as, to edit the boot line, change parameters etc.. If you press "e" (different on different distros, but it may be the same in Zenwalk) you can go to the line that starts with "kernel", and, after leaving a space write xvideo=vesa at the end of the line, then press "enter" or F10 to boot the kernel with the new vesa option. It may even help with the booting of the Zenwalk distribution you are using.
Hint: When you see the Grub screen (The one with the kernel choices, or it may have a memtest option, or a safe booting option), if you press the down arrow, then the up arrow, it will suspend grub and let you read what the options are. You can then add the vesa option and press the option to continue to boot.
BTW - If this all seems confusing, it is to me also. I don't have a Zenwalk install in front of me, so, I'm just passing ideas. If you have any specific questions after looking for the grub screen and can tell us what you find, I may be able to give you some more specific instructions. I'm sorry, but I looked at the Zenwalk site, and, couldn't find any pictures that would show me what their default grub screen looks like. If this seems to be too complicated for you as a beginner, you may just want to download AntiX and start from scratch.