Short answer
Each follows a different lunar rule. Pradosham is the 13th lunar day (Trayodashi) in the evening twilight around sunset, and comes twice a month. Sankashti is the 4th lunar day of the waning fortnight, kept on the day it is running at moonrise. Amavasya is the new-Moon lunar day, once a month. The Festivals tool lists this month's days for your city.
How ChiChakra reads this
ChiChakra checks each rule against local times in your city: sunset for Pradosham, moonrise for Sankashti, and the running tithi for Amavasya. That is why the same observance can land on different civil days in Chennai, London and Chicago in the same week.
Planning the month
Pradosham: the puja belongs to the twilight around sunset, so plan the evening rather than the whole day. When it falls on a Saturday it is called Shani Pradosham, and many families give it extra importance.
Sankashti: the fast usually ends after you see the Moon, so check your local moonrise. If clouds hide the Moon, families commonly go by the calculated moonrise time.
Amavasya: a day for remembering ancestors with Tarpanam, and traditionally a quieter day rather than one for new starts.
Good to know
ChiChakra uses the Sun's centre with no refraction for sunrise and sunset, so its times can differ from a printed panchang by a few minutes.