Wipe regularly with damp cloth. The best way to keep a stainless steel stove from getting out of hand is to wipe it down with a damp cloth after each use. This way you catch the scrambled eggs, gravy and soup splatters before they dry and become a nuisance.
Use soap for heavier cleaning. Regular old dish soap works well for cleaning off the eggs, gravy or soup splatters you did not get to right away. Saturate a rag or sponge, wipe down the stove and rinse well. You may want to leave the soap on a for a few minutes to soften up any caked-on food.
Scrape lightly if necessary. If the sponge saturated in soap just isn't cutting it, you can gently, lightly and very tenderly scrape off any baked-on stuff. Never use metal to do this but rather pick something that won't scratch, like a fingernail, old credit card or plastic or rubber spatula. Do not scrape into the stove top but gently drag the scraping item along the top.
Break out the big guns. Get a stainless steel cleaner, a specially made formula to spruce up stainless steel. These are sold at hardware and home stores (see Resources).
Wipe dry. Whatever cleaning method you choose, make sure to wipe the stovetop dry with a soft cloth or even newspaper to avoid unseemly streaks.