When aminals were sacrificed, either prior to the Golden Calf incident, when performed privately by persons such as Noah or Abraham, or for that matter, during the temple period, what kinds of prayers or incantations accompanied the sacrifices?
There were many different types of sacrifices offered, at different times in history (e.g. by the Patriarchs, in the time of the Tabernacle and the Temple), each with different intentions and associated laws. The early sacrifices were all offered as thanks to G-d, from Adam, Cain and Abel, through Noah, to the Patriarchs. I don’t know what words were said, but the intention was to thank G-d for salvation or promises and covenants for the future. The animals were slaughtered and entirely burnt on an altar.
In the times of the Tabernacle and the Temple these kinds of sacrifices continued, in addition to sin offerings, which would involve a confession prior to the slaughtering, and this animal was entirely burnt on the altar. There were also mandatory daily, weekly and festival offerings which were offered by the Cohanim on behalf of the entire nation to the accompaniment of the singing of the Levites. These were also entirely consumed by the fire on the altar. The Passover sacrifice had many rituals associated with it, most of which we do today at the Seder (e.g. eating as a family group, telling about the exodus and eating matza and maror). Certain specific sacrifices had other conected rituals (e.g. a Nazir had to shave his hair at the time of the sacrifice, a metzora had to go through a specific purification process etc.). These can be found primarily in the book of Leviticus.
I hope this answers your question.
With best wishes
Rabbi David Sedley