I have a Victorinox Swiss Army Classic multi-tool. What do you recommend for sharpening the blade and scissors?

That depends entirely on your skill level. You can use a ceramic or diamond rod, or a wet stone. Or some fancy angle keeping deal.

like the Spyderco Sharpmaker. It will sharpen your Classic and just about everything else in sight. I was a whetstone guy for years and years, but I pretty much do every thing on the Sharpmaker now.

Yep, that would be a good example.

Which Spyderco Sharpmaker? The one(s) I've seen online is/are around 4-5 times the cost of my knife.

Do a google search for Lansky. Select the "Standard Sharpening System Kit" in the shopping results. You can get one for around $20-$25 that will put a razor edge on just about anything.

I just looked online and it seems to go for $55 now, up a noticeable amount from what I paid a few years ago. Between the kitchen, the workshop, and my outdoor stuff, the Sharpmaker will do a very fine job on just about everything - there is a special setup just for scissors, for example.

Something that would be a bit more appropriate for your situation would be a medium diamond sharpening steel and any of several fine grade hones. With a bit of practice, you can get a keen edge and you will have something you can carry in the field, a place where I do not carry my Sharpmaker.

I use one of those white rods on my knives. (Is that ceramic?) I've also used it on scissors, but not tiny ones. I suppose it'd work good for those. And I've got a small whetstone that I use too. The blades are always sharper than when I started, but I never get them razor sharp.

finallyME might have hit on why

??? All the spyderco sharpmakers I found online were around $60. All Victorinox multitools are around $60 or more. Are you sure you are talking about a multitool, or a pocket knife?