Basic rule is that a כלי ראשון can cook but a כלי שני cannot. To illustrate, beis Hillel says that one can pour some cold water into hot water in a כלי שני. Beis Shammai, however, is stringent and does not allow it unless the hot water is being poured into cold.

Even though beis Hillel permits cold water to be poured into hot water in a כלי שני, he is stringent if the hot water is in a bathtub. (Presumably, the temperature there is hotter than drinking water) A washbasin is not included in this stringency.

Most spices may not be added to a כלי ראשון but may be added to a כלי שני. There are 3 opinions regarding salt. It’s either exactly like regular spices, or it may be added even to a כלי ראשון, or it may not be added even to a כלי שני.

Oil that drips from a burning lamp is muktzeh and cannot be moved. One should not move a non muktzeh item for the sake of a muktzeh item, and so, it is forbidden to bring a cup to catch the falling oil.

Even though one generally cannot move a non muktzeh item for the sake of a muktzeh item, the rabbis were lenient for cases of saving things. For example, a chicken lays an egg on Shabbos (muktzeh) one can cover the egg so people don’t step on it and crush it. This is only for frequent, regular cases but not for anomalous cases, like an egg on an incline one can’t put out a kli to catch the egg.