Get your water boiling.
Dishes can be made of several substances: china, earthenware, glass, wood, various plastics, metal. Some can be kashered [made kosher], some cannot, some are debatable.
Leviticus 6:21 says, "And the earthenware vessel in which it was cooked shall be broken; and if it was cooked in a brazen vessel, it shall be scoured, and rinsed in water." This verse is the basis for certain rules of kashering dishes, and what may or may not be kasherable.
Whether or not you can keep the dishes you already own depends on the material from which they are made. If the dishes are "earthen," that is, china or stoneware, it is most likely you will not be able to keep them, at least not for immediate use. Unglazed earthenware cannot be kashered at all, as the porous ceramic permanently absorbs juices and flavors from foods.
Nowadays, glazed chinaware can be relatively inexpensive, especially compared to 50 or 150 years ago. Consequently, Orthodox and Conservative opinions on keeping dishes that were previously used in unkosher kitchens have become stricter. The general opinion is, chinaware cannot be kashered.
Thickly glazed china or fine china, family heirloom material, can sometimes be made kosher, but it must sit, unused, for at least 12 months; have a rabbi check your china to determine if this rule is applicable. This is the amount of time deemed necessary for dissipation of all unkosher flavors. If the glaze is thin and could easily flake off, the dish likely cannot be kashered. The alternative is to find a kiln and heat the ceramic items at kiln temperatures (about 2,000 degrees Farenheit), but at that heat the china might break.
The rule regarding chinaware applies not only to place settings, but to serving platters, as well as mixing bowls, fruit bowls, and so on. Items that are never used with hot food, such as a ceramic fruit bowl or a sugar bowl, are not treif [unkosher] and don't need kashering.
Teapots, if they were never used with flavored tea, may not need to be kashered and can be kept; check with your rabbi.
Glass has been controversial, in both Conservative and Orthodox rulings, in terms of determining what material category it fails under. Glass is made of sand, so is it earthenware? Yet it does not absorb as earthenware does, so is it like metal? In the Shulhan Arukh, Joseph Caro writes that glass dishes "need no kashering since they do not absorb," while Moshe Isserles writes, "And some are more stingent and say that even purging by boiling is of no effect in their case." The ultimate Conservative and more common Orthodox conclusion is that glass is neutral, and can be kashered simply by washing.