You'll get the first one from the event "P School Suggestion" when your prosperity reaches 80.
After the first school, you need to trigger an event chain for any others you want:
1. Maintain more than 50,000 credits of liquidity after buying at least two students from the school of your choice at the market. Eventually "RES Endowment" will trigger for that school, asking for money. Do not choose "Make a generous contribution" in this event (or "Patronize them without taking sides", for TFS); those options award cheap reputation without providing a permanent endowment for the school, and you need that endowment to trigger the second event.
2. After a school is endowed, "RES Move" may trigger for the same school, where you'll receive a personal call from the head of the school asking to open a branch campus (or subsidiary lab/pasture/futa community, for GRI/TCR/TFS respectively). Accept the offer and you'll have a shiny new branch!
"P School Suggestion" is a plot event and will always trigger once (assuming you meet the prosperity requirement), so the first school is more or less guaranteed. "RES Endowment" and "RES Move" are both nonindividual random events and may take some time to trigger (or, if you're unlucky or qualify for a very large number of NREs, may never trigger at all). You can improve your chances for "RES Endowment" by buying more slaves from the school in question.
A lot of people miss the "buy two students at the market" requirement for "RES Endowment", so if you're having trouble, make sure you've done that.