Answers by Rabbi Chaim Hillel Raskin:
Halacha records a prohibition to eat meat with fish as it is harmful to one’s health. Poskim write that it applies to chicken as well, even if the two foods weren’t cooked together.
Does a small amount of fish that fell into a meat dish, or vice versa, become nullified?
While some are of the opinion that a harmful ingredient cannot be nullified, the consensus of poskim rule that it is no worse than non-kosher food which is nullified (botel) in sixty times another food.
Can I cook fish and meat in the same oven?
One may use the same oven for fish and meat provided that the oven is clean of drippings. To cook them at the same time, one of the dishes must be covered, so that the aroma (reicha) of one doesn’t penetrate the other. If both dishes were not covered, it is permissible b’dieved.
Can I use a meat pot to cook fish?
The widespread custom is to be lenient with using clean dishes which contain only residual flavor in the dish. Some are mehader to have separate pots (but not other dishes), since cooking draws out the flavor more.
Likewise, it is permissible to cook an onion that was diced with a fleishig knife together with fish, however it is preferable to use one that hasn’t been used for meat in 24 hours.
While according to halacha they may both be on the table at the same time, the custom is to remove the fish before bringing out the meat as an extra precaution.
Between eating fish and meat one should make sure his hands are clean and rinse his mouth, (many have the custom to say a “l’chaim”), and some also rinse their fingers.
(Source: Markaz Anash)