This answer is relevant in case cell lines and not much for primary cultures
When I first started doing cell culture, I used to count the cells before passaging. The idea is to seed enough cells to keep the population viable but not so much that you have to re-passage within 2 days. However as I became good with the technique I stopped counting as I would have a decent estimate of how much to seed. For seeding cells for some experiments (transfection/drug treatment etc) it is crucial to count the cells.
How much to seed depends on what the division time of the cell type is and how much you need after a given duration of time. You have to calculate it accordingly. For example a fully confluent well of a 6-well plate will approximately have 1 million Hela cells (this number would vary for different cell types because of the difference in sizes). Hela cells have division time of ~17h (https://biology.stackexchange.com/a/13724/3340) If you need ~70-80% confluency after 17h then you should seed ~0.3 million cells.
Likewise, a T-25 flask would have ~3 million cells in full confluency. So, if you need a confluent flask in 3 days you can calculate how much cells to seed.
See this table in Nunc website for number of cells at confluency and optimal seeding density for different culture vessels.
Note that some cells (mostly primary cultures) may need to be seeded at high densities because they may need contact with other cells for survival. So, I cannot give any authoritative answer for primary cultures. Moreover, since it depends on the cell type, all examples cannot be listed here.