The order followed in the Temple:[1]

1. Passing the blood: There was a line of Kohanim standing who would pass the vessel of blood from one to the other until it reaches the hands of the Kohen closest to the altar, who would then sprinkle it one time on the foundation of the altar. The Kohen would then return the empty vessel through the line of people, until it reached the Kohen performing the slaughtering and once again became refilled. The Kohen first takes the full cup and only then takes the empty cup.
2. Preparing the animal: After the blood is sprinkled, the animal is completely skinned and torn open. One presses its innards until all of its feces comes out. The limbs to be offered are then removed. The limbs are then placed in a Temple vessel and salted. The Kohen then offers each limb onto the altar. Each limb is offered individually.
3. Hallel: Hallel is recited [by the Levi’im] while the animals are being slaughtered and offered to the altar. It is repeated until all the people in that group have completed their sacrifices.
4. Cleaning the Temple: After all the sacrifices are brought, the Temple floor is washed from the blood. This applies even on Shabbos.

_____________________________

[1] Admur in Seder Karban Pesach; Regarding the training of the Kohanim-See Karban Pesach Kehilchaso 2-4; Regarding all other matters of the offering: See Karban Pesach Kehilchaso 31-87