The mouthpiece is likely silver plated; silver polish will work well on that, as would the aluminum foil trick. (Since it's silver plated as opposed to solid silver, perhaps try that trick first.)
I think the aluminum foil trick: http://scifun.chem.wisc.edu/homeexpts/tarnish.html will likely be the least damaging, but I'm not an expert in metallurgy.
The slides could be silver plate, or (in my opinion more likely) nickel plate. My nickel plated trombone bell didn't tarnish as "dark" as my solid silver one. It was more of a brassy look, likely because of the copper in the nickel silver. I can't say you'll always be able to tell the difference… but it's a shot.
Nickel silver can usually be cleaned and polish like silver; all mine have been fine with that, as it's designed to be a silver look-alike. Check out this resource: http://www.ehow.com/how_5210485_clean-nickel-silver.html.
I don't personally know about toothpaste, as I haven't heard of it's use. I'd imagine it'd put a slight patina of small scratches on any polished metal, as it's an abrasive.
Whichever technique you ultimately use, clean all pieces thoroughly before reassembling and using.