The most common route from graduating in Germany to permanent residency is the EU Blue Card. Once you have a qualifying skilled job offer at or above the current Blue Card salary threshold, you can apply during your 18-month post-study permit.
Blue Card thresholds are set annually, with a lower rate for recent graduates (degree less than three years old), shortage occupations, and qualifying IT specialists. Only guaranteed fixed gross salary counts. Bonuses, stock, and commissions don't count.
With a Blue Card, you can apply for permanent settlement (Niederlassungserlaubnis) after a qualifying period, provided you've made pension contributions across it. The qualifying period is shorter if you have B1 German than if you only have A1. This is one of the fastest routes to permanent status in Europe.
If you don't qualify for the Blue Card, there's also a graduate-specific route under §18c AufenthG that leads to Niederlassungserlaubnis after qualifying work and B1 German.