With Zills that are nice and shiny when new, you have two choices:
1) Polish them frequently,
2) Let them go
au nateral.
If you want to keep them shiny, you can polish them with brass/copper polish, or car wax. Both leave somewhat of a protective coating. Always, always, always wipe your shiny Zills down with a soft polishing cloth before putting them away - and don't touch them with your bare fingers after you wipe them down. Then keep them in a fairly airtight container - the little bags that Saroyan sells (or used to anyway) are pretty airtight, a ziploc plastic bag is very airtight, but not particularly glamorous.
If you keep them in an open weave bag that admits air, they will tarnish very quickly.
I have three sets - one set I keep polished, one I let tarnish, and the third are cast bronze
Zildjians which you couldn't polish if you wanted to (rough castings).
As for playing them, I heartily recommend:
Zills on Fire Finger Cymbal Instruction--zils, belly dance, drum rhythm lessons