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First you should know that stainless steel usually contains some iron so will rust and it is not stainless.Stainless steel varies in quality, and manufacturers may use different grades for different parts of the grill.
Stainless Steel Cleaning - Weber Grills
How to Protect a Stainless Steel Grill from rusting | 9mmSteve
2 coats of car wax followed by olive oil.
You need to wipe most of the olive oil off or you will get brown spots where there is excessive oil.
Repeat a couple of times each year.
How to Prevent Rust on Your Grill Ghar-Broil
Stainless Steel Care and Cleaning
Spray a small amount of WD-40 or Sheila Shine on a
piece of paper towel and rub with the grain over the
surface of the appliance. Using a clean terry cloth or
paper towel, polish the appliance .
Furniture polish, such as Old English, can be effective
for polishing stainless steel and protecting the
surface.
Clean Your BBQ! - Clean My Space
Inside the lid:
remove the build-up regularly with a non-scratching brush (stainless steel), non-scratching scraper or non-scratching sponge.   Once you’ve scraped 
everything off (and let it fall to the bottom of the cookbox), wash the interior lid with a soft sponge, warm soapy water, then rinse and buff dry with a soft cloth.
The Bottom of the Cookbox:
Remove as many components as you can, safely.  Use a non-scratching scraper to remove any carbonized grease, build up or fallen food. 
 Many have pull-out trays that can be easily emptied into a garbage).  This can be wiped clean occasionally as well, following the same protocol as above; clean with warm soapy water and a non-scratch sponge to remove grease, and boot up to a degreaser.
 
 Tools:
 fill your sink with warm water.  Add a teaspoon of dish soap and a half cup of baking soda.  Soak the tools or accessories and then scrub clean with a non-scratching sponge. 
Links:
How to Clean a Stainless Steel Grill - Consumer Reports
How to Maintain Your Gas Grill - Consumer Reports
Cleaning Video Consumer Reports