Question: What do I need to do between eating fish and meat?
Answer: While the Gemara (Pesachim 76b) only warns against the dangers of eating meat that has been cooked together with fish, the Shulchan Aruch (YD 116:2) extends this prohibition to eating fish with meat or chicken (Pischei Teshuvah YD 116:2) together.
As this danger seems to conflict with contemporary medical knowledge, the poskim debate as to whether this applies nowadays.
The Mishna Berura (173:3) and Aruch Hashulchan (YD 116:10) quote the Magen Avraham (OC 173:1) who writes that this is one of many natural phenomena mentioned in the Gemara that no longer apply. The Chasam Sofer (YD 101) suggests that this may be why there is no mention of this in Rambam.
Nonetheless, the Chasam Sofer concedes with the majority of poskim (Chochmas Adam 68:1; Maharam Shick YD 244; Kaf Hachaim OC 173:9), who write that this prohibition still applies even if we don’t understand the danger (See Yad Ephraim 116:3).
The Tur (YD 116) writes that his father would wash his hands and eat wine soaked bread to cleanse his palate between fish and meat. This wouldn’t apply to one who uses a fork to eat their fish (See Kaf Hachaim ibid). Most ashkenazim follow the Darkei Moshe (OC 116:3) who writes that there is no need to wash one’s hands at all. Many sefardim are particular to rinse their mouths and hands in between (Kaf Hachaim OC 173:4; Yalkut Yosef 173:2).
The Chochmas Adam (68:1) writes that one should drink something in between though the Rema (OC 116:3) and the Mishna Berura (173:4) writes that one should both eat and drink something.Many are particular to have a stronger drink, rather than water in between (See Shearim Metzuyanim Behalacha 33:2).