A rider and their mount share a single character's set of actions (one standard, move and minor action, one immediate action between turns). The player may use each action on either the rider or the mount. Most commonly, you'll have the mount take a move action and the rider take a standard action to attack.
Importantly, though: the mount and rider are treated as two separate creatures, and actions taken by the mount are the mount's actions.
Do the rider's conditions (e.g. slow) apply to the mount?
No. The mount and rider have separate conditions. The mount can still move at full speed even if its rider's slowed.
There are a few exceptions, like how if the mount is squeezing, the rider is too, but they exist where it makes sense and are few.
What if the rider's forced to move?
If the rider is affected by a power that requires it to move (eg: "The target must move its speed away or take 10 psychic damage") can it move its mount to qualify? Does "its speed" apply to the rider or the mount?
It'd depend on what the power itself says, but the movement of the mount is not the movement of the rider. In that case, the rider might have to dismount and move on their own.
What if the rider's immobilized/restrained/grabbed?
Immobilized (RC p232): The creature can't move, unless it teleports or is pulled, pushed or slid.
Fine, the rider can't move, but the mount beneath them can, and that mount can take its rider with it.
Grabbed makes a character immobilized, so the same again applies. Plus, from the Grabbed condition's rules text:
This condition ends immediately on the creature if (...) the creature ends up outside the range of the grabbing power or effect.
A mount could tug its rider free of the grip of an orc, or make steady progress trying to pull its rider free of some very long tentacles with a ranged grab effect.
Restrained (RC p234): The creature can't move, unless it teleports. It can't even be pulled, pushed or slid. The creature takes a -2 penalty to attack rolls. The creature grants combat advantage.
In this case, the rider cannot be moved. Restrained conditions are about simply being immovable from the spot you're in, unless you magically teleport out of that mess. Generally, restrained characters are caught in vines, tentacles, spiderweb, manacles attached to a wall, or a monster's jaws.
Can the mount still move away? The rules are silent: make a judgement call. If it can, the rider would be dismounted and left where they are.