Many people have asked me how to polish silver instruments. You need to do two things. First you need to clean and polish the horn, and second you need to keep the tarnish from coming back very soon.
Silver is a relatively soft plating, so you should NOT use a standard metal polish. Use something that is made for fine silver finishes. I prefer Hagerty Silver Polish with Tarnish Preventative. The kind I use most often is a cream in a bottle. You put some on a soft cloth, polish an area of the horn, let it dry for a minute or so, then polish it off with a clean soft cloth. It leaves a protective coating that will help keep tarnish from forming.
For hard-to-get-at areas, Hagerty also makes a spray polish. It is easy to spray it between valves and on other tight spaces. Then you can thread a clean cloth between the valves to pull back and forth, which will polish it off.
Of course there are many Hagerty silver polish products on Amazon:
Hagerty Silver Polish
3M makes a polish for silver called Tarni-Shield, but it doesn't give as nice a shine as Hagerty in my opinion.