I used to use a product called Sheila Shine which makes stainless steel shine and
it also hides fingerprints. It leaves a thin coating to keep the surface shiny. It always felt a little oily to me, but it gives a nice shine and the oil feel dries and goes away. On the recommendation of the dealer when we purchased a stainless steel cooktop and oven for our cottage, I've switched to a 3M product in a pressurized can. I don't have it here, and I can't remember the name of it, but it works great, and also leaves a thin protective coating that doesn't feel oily. The coating protects the surface (just like Orchid described) so you don't have to clean it as often.
The other thing I learned about a stainless steel cooktop is that you can clean it with oven cleaner. Again I learned this from the appliance dealer. Use the fume free oven cleaner, and don't use it if your cooktop has a coating. I had been struggling with stains and burned on grease for a long time before I tried the oven cleaner. I tried it on a tiny spot first and left it for a half hour. Wow, it really works. The grease and stains just wiped off. My cooktop looks like new now.
Happy Spring Cleaning.