After eating dairy food (with your hands) you must wash your hands with water before eating meaty food. (ShA 1 – Rama 3 – MB 2)
This washing does not require a keli. It must reach only up till the second knuckle, and if something blocks the water from contact with your skin, it’s okay. However, you need to dry your hands before resuming your meal. (ShA 3 – MB 6)
These days it is not so dangerous to eat meat and then fish without washing in between. Even still, you should eat and drink something in between. (MB 3,4)