For general polishing chores, I use polishing cream (like Goddard's, Wright's or Connoiseurs). These do an excellent job, even on heavy tarnish, but leave the nooks and crannies with their characteristic patina. The creams are also not abrasive and can be used even on sensitive surfaces. But you should always use care when polishing items that have enamel or soft stones, as these can be easily damaged. For light polishing and touch up polishing I use a silver polishing cloth. (My favorite is Goddard's.)

I avoid dips and active chemical cleaners except on the truly horrible surfaces. I do use dips occasionally for a light, quick polish on irregular surfaces, especially jewelry. But do so with care.  DO NOT USE dips on enamels, soft stones, organics (pearls, etc.), filigree wire work or costume jewelry. I have not tried any of the electrolytic or home brew  methods, but have seen the disastrous results of such things as toothpaste and lemon juice concoctions. Mechanical polishing with polishing compounds is best left to the pros.