It is ossur to fast on Shabbos and Yom Tov (except for Rosh Hashana). Make sure you eat before chatzos even if you don’t plan on fasting. (ShA 1 – MB 1,2)
Some poskim are lenient if the delay is due to learning Torah and daavening. (MB 2)
A person harmed by eating is allowed to fast on Shabbos – to him fasting is his oneg. If eating a little bit doesn’t harm him, he must eat that little bit. (ShA 2, 3 – BeH “tzaar”)
Teshuva fasting is ossur. (MB 5)
You are allowed to cry from feelings of d’veykus. Some poskim also allow crying in order to feel better. (MB 4)
Achronim write that you are allowed to supplicate on Shabbos “Yehi ratzon sh’lo echta oid…”. The prohibition of requesting includes only about the sick or about parnassa, the likes which have to do with tzaar. But regret about your sins should be expressed every day as long as it isn’t in the form of Vidui. (MB 22)
One who is suffering great tzoros is allowed to beg for mercy and fall down on his face in private, but fasting is ossur. (ShA 8 – MB 26)
We may cry out and beg in prayer on Shabbos in case of an emergency — even for an individual. (ShA 9)
You may bless a dangerously ill person to get better in the normal manner on Shabbos. The same applies for a women in childbirth or the week after birth. However, when making a Mi Shebayrach for a patient who is not in danger, we change the nussach to “… Shabbos hee miliz’ok urfuah k’rova lavoh”. (Rama 10 – MB 28)