Planets form from a protoplanetary disk that has been rotating around its star. The initial energy that makes them rotate really matters to me.
Two rocks placed in space with no relative motion are going to be attracted by gravity, and hit. 3 rocks, placed in space with no carefully rigged symmetry, will likely miss each other, as the gravitational attraction of the additional rock changes their course. Those near misses are the beginning of rotation. Multiply that effect by trillions, and you have a protoplanetary disk. Of course, a lot of those molecules, dust and rock won't even start out with zero relative velocity, so near misses are inevitable.
The prior gas cloud will start with some small random motions and density variations left over from its formation. As a patch starts to contract any small amount of rotation gets amplified as the cloud collapses, because of the conservation of angular momentum.
The answer is turbulence. Stars are formed from large gas clouds that collapse under their own weight. As they do so, they become unstable to fragmentation and break up into smaller collapsing pieces. Turbulence in the original cloud means that each of these pieces has its own individual angular momentum and rotational energy, even if the total angular momentum of the original cloud was zero. The original turbulence can be injected in many ways, including gravitational interactions and collisions with other clouds, the tidal effects of the galaxy, shocks from supernovae, the winds of massive stars, etc.
As the collapse proceeds, dissipative interactions in the collapsing cloud cause it to radiate away energy, but very little angular momentum. Both quantities must be conserved, so the system tends towards a configuration with minimum energy for a fixed amount of angular momentum - which is a rotating disk.
In order to accrete into the central protostar, the material in the disk has to somehow lose angular momentum in order to fall inwards. It does this by transferring angular momentum outwards through various viscous processes. Ultimately though, some of the disk material has to escape with the angular momentum to allow some of the material to accrete into the star. There are a number of processes that can achieve this including "disk winds" and photoevaporation.
Other answers good, but this question left behind:
Why isn't all the disk material been swallowed by the star?
Because of the excess of angular momentum carried by some particles. They insist on orbiting the star instead of falling directly to it.
There is a complex process called accretion. The disk constantly transfers angular momentum to the outside, material to the inside and thermal radiation away. In most cases, some material is also blown away in polar jets.
The physics is the same, no matter if it is a galaxy, a protoplanetary disk, a shiny disk around a neutron star or a black hole, or something smaller like a planetary ring. Well, the scales differ.
There are three catches:
For the accretion to work, one needs some friction between the concentric rings of material. Easy when you have dense enough gas or plasma and stops working when you don't have gas anymore. Dust works somehow, large rocks (planets, moons) - not really.
Some local disturbances in the accretion process may form local mini-accretions in the disk. They form planets/moons that "steal" some material from the star.
Light - once the star is bright enough to reach Eddington limit, the light pressure over the accreting gas overcomes the gravity and the gas gets blown away. The star stops growing and its planets stop growing, too. The dust leftovers get gravitationally stirred until most particles either coalesce over the star and the planets, or get an escape orbit and, well, escape.
Answers, in order:
Why did the protoplanetary disk start rotating around the star?
Star and disk form around the same time.
The rotation forms because all other orbits will cross the orbit of the disk and collide. Put another way, the rotation disk is the leftover movement of particles after all symmetrical factors have cancelled each other out in collisions (inelastic ones, elastic ones do not average out anything).
Where did the initial energy to rotate come from?
From the potential energy of the particles that were far away from the center of gravity when the dust/gas could started collapsing.
This also explains why the disk is rotating faster than the original dust cloud: As masses go to the center of a rotation, they rotate faster (conserve angular momentum).
Why isn't all the disk material been swallowed by the star?
I couldn't answer that better than @fraxinus has.
Summarizing/adding to previous specific answers:
Why did the protoplanetary disk start rotating around the star?
The disc forms because it was rotating. Without net rotation it wouldn’t form a disc... it will stay a spherical symmetric cloud.
Where did the initial energy to rotate come from?
A small angular momentum NOT energy is required the “angular energy” gets magnified during gravitational infall while conserving angular momentum like the figure skater spins faster pulling arms in with small initial rotation.
Why hasn't all the disk material been swallowed by the star?
As it speeds up it is basically orbiting faster and faster until centrifugal force balances gravity. Equilibrium.
