What yields the best edge, a stone or the double wheel device?
I personally prefer the stone, it gives you more control and flexibility compared to the double wheel.
I don't have the confidence to do a good job, so make a relationship with a good local butcher and ask him to sharpen for you - with a suitable gratuity (and buy some steaks too).
I say get comfortable with a whetstone. It's a cheap investment and with practice, allows you to sharpen your own knives whenever you want. Double wheel grinders grind down knives way too much. In a sense, they aren't actually sharpening your knives, just creating a new edge. Stones sharpen the edge that's already there. Your knives, and the edge, will last a lot longer; it puts much less wear and tear on the knife. And, as mentioned, double wheel grinders require a lot of balance; you have a lot less control than with a stone. There's also the chance of getting an uneven edge unless you keep a very steady rhythm as you move the knife along the wheel. Plus, without experience, it can be very dangerous for many reasons.
Similar to your butcher, you can check with a chef. Many chefs have knife sharpening services that come by and sharpen all the kitchen knives. You could bring yours by and have them done at the same time I personally prefer a whetstone for my sharpening. Sit down to watch the news and spend some time sharpening.
I just use a sharpening rod. Here's a good, simple how-to video on You Tube:
My local cooking supply store and also hardware store offer knife and scissor sharpening service for a modest fee, like once a month. (In case you don't have a nearby butcher or chef you feel comfortable approaching.)
I have a very large knives so I use a variety of methods. I have a professional electric model I use for larger chef's knives. I also have a smaller electric model from Shun that is dedicated to Japanese knives which have different bevel. For smaller utility and paring knives I still use the old fashioned whet stone. I keep a steel handy too but that doesn't really sharpen it just hones.
oops, that should read "...large collection of knives".
Sharpening a good high carbon kitchen knife is Like Zen....So many branches to follow Grasshopper! My method is using a ceramic rod and a light steel that only need to be dressed up. If edge is really dull, then incorporate a diamond rod to bring back the edge. The sharpening motion, I use is a little different than most on the rods. I draw the edge backwards, away from me on the rods. This eliminates the burr, you'll get from drawing the sharpe edge forward away from you on the rods. On slightly dull good edged blades, use ceramic, than steel. On duller edges...use diamond first, then ceramic, then steel. Finish off with drawing away from you on cardboard to polish edge, if you want to. You could use a two sided diamond rod, then steel and not buy a ceramic rod too. Doesn't save money, just less rods to use, since diamond rod has two grits! Either combination works great!
i started with some relatively cheap ikea santoku knives several years ago and am now spending more on some victornox knives. cheap doesn't mean they don't work well. anyway, before i started spending more i read the article that i'll include at bottom of my post. it gave me the confidence to use a sharpening oilstone. i bought a relatively inexpensive one on amazon along with the oil have been really pleased with the results. the article is pretty long but really a great resource - hence i book marked it. there is a second continuing article but you can find that one yourself.