Is it true that stars start with a rocky core, then collect hydrogen by the gravity of that rocky core, just like planets collecting materials?
No, stars do not have a rocky core. They start as clouds of gas, become gravitationally unstable and collapse.
Some planets are thought to start this way too - though probably not the ones in our solar system, because the outer three three giant planets have solid cores and Jupiter is likely to also. The two modes of giant planet formation are reviewed in detail by D'Angelo et al. (2010).