The Shulchan Aruch (Yoreh Dei'a 116:2) rules that one must ensure not to eat meat with fish, as doing so poses a medical risk. Specifically, the Shulchan Aruch mentions that eating meat and fish together increases the risk of contracting Tzara'at (a type of skin disease).
Therefore, one who eats fish must first rinse his mouth before eating meat, and vice-versa. Likewise, one must ensure to use different dishes and cutlery for fish and meat. At a buffet, for example, one must make a point of using different utensils and silverware for fish and for meat.
When it comes to the prohibition against eating milk with meat, Halacha establishes that two friends or acquaintances may not eat together on the same table if one eats meat and the other milk. The Rabbis were concerned that the two might share each other's food, and they therefore forbade two friends or acquaintances from eating meat and milk together at the same table. When it comes to fish and meat, however, Chacham Ovadia Yosef rules (in Yabia Omer, vol. 6, Yoreh Dei'a section, 9) that this provision does not apply, and two friends or acquaintances may eat fish and meat at the same table. He explains that since eating fish and meat together is forbidden due to a health concern, people exercise particular care in this regard, and so it is unlikely that a person eating fish will forgetfully take some of his friend's meat, or vice-versa. (This ruling also appears in Chacham Yitzchak Yosef's work Issur Ve'heter, p. 70, Halacha 2.)
It is permissible to cook fish in a pot that had been used for cooking meat, or vice-versa. So long as the pot is clean and has no actual fish or meat on its surface, one may use it for cooking either fish or meat, even if the pot had been used with the other less than twenty-four hours earlier. See Taz (commentary on Shulchan Aruch written by Rabbeinu David HaLevi 1586-1667) on Yore Deah, Siman 95, Seif Kattan 3.)
Summary: One may not eat meat and fish together; after eating meat or fish, one must rinse his mouth before partaking of the other, and he must ensure to use different utensils and cutlery. Nevertheless, two people may sit together at the same table, one eating fish and the other meat. A utensil that had been used for cooking fish may be used for cooking meat, and vice-versa, provided that it is clean.
