Without seeing the objects it’s hard to give concrete advice in regards to cleaning, but I can offer a few ideas for storage and maintenance that I hope will be helpful. I’d encourage you to talk to a local conservator who will be able to give you a better idea of the condition of your collection and the types of treatments that it might need.
For most solid silver objects, regular, gentle polishing with a light abrasive polishing cloth will be enough to brighten up the surface and remove some light tarnishing. However, polishing silver-plated objects with these cloths can lead to damage and removal of the plating, so it’s important to know the collection. Polishing compounds (useful in removing severe tarnishing or pitting) can become stuck in crevices of intricate surfaces and may damage the abalone or other settings you describe, so I’d recommend against using them on your collection, and would advise you to consult with a conservator about any objects that require further treatment.
For storage, silver should be kept in sealed storage away from sulfide and chloride gases, which can lead to tarnishing. I’d recommend that you avoid wood cases unless they are *thoroughly* sealed, since they can off-gas, leading to more tarnishing. Silver objects can be wrapped in acid free tissue paper or (even better) silvercloth and placed in sealed polyethylene bags, or archival boxes. Silvercloth is a scavenger that will effectively prevent against tarnish. It is quite effective, but will need to be replaced periodically.
This link (while a little bit old) may help to fill in a few of the gaps I missed:
http://www.bishopmuseum.org/wp-content/uploads/2016/04/cnsv-silver.pdf