Although it is forbidden to make preparations on Shabbat that are needed only after Shabbat, it is permissible to make preparations for later on Shabbat. And thus while it is forbidden on Shabbat to wash dirty dishes that will be needed after Shabbat, one may wash dishes that are needed later during Shabbat. For example, after Friday night dinner one may wash dishes and cutlery that will be needed for Shabbat lunch, and after Shabbat lunch one may wash dishes for Se’uda Shelishit. As Se’uda Shelishit is generally completed soon before the conclusion of Shabbat, it would not be permissible to wash dishes after Se’uda Shelishit, though if one does plan on eating again after Se’uda Shelishit before the end of Shabbat, then he may wash the dishes. Cups and glasses may be washed because people often need to drink, and therefore even after Se’uda Shelishit it is permissible to wash drinking utensils. These rulings are codified by Hacham Ovadia Yosef in Hazon Ovadia (vol. 6, p. 100; listen to audio recording for precise citation).
Hacham Ovadia also cites the ruling of Rav Moshe Feinstein (Iggerot Moshe, Orah Haim, Siman 74, p. 145) that if one wants to put dirty dishes in the dishwasher after Se’uda Shelishit for the sake of orderliness, so the sinks will be clean, he may, as long as he places them in randomly, and does not set them up to be washed after Shabbat.
It is forbidden to use sponges on Shabbat, and using them may even violate a Torah prohibition, as using them invariably entails extracting absorbed liquid (Sehita). It is permissible, however, to use a plastic sponge that does not absorb water. It is also permissible to wear dishwashing gloves while washing dishes on Shabbat, since they do not absorb water.
Summary: It is permissible on Shabbat to wash dishes that will be needed later on Shabbat, but not dishes that will be needed only after Shabbat. One may load dishes into the dishwasher on Shabbat if he does so randomly, and not in an organized fashion preparing for the dishwashing cycle that will be run after Shabbat. It is forbidden to use absorbent sponges on Shabbat, but non-absorbent sponges and dishwashing gloves may be used.
