How can I download photos from Photo Stream onto a new iPad? I bought a new iPad and I want to download all my photos from Photo Stream there. How do I do that?

Currently, it only syncs newly taken photos, but not all those taken in the past which are available in Photo Stream through my other devices.
Photostream isn't really a synchronise mechanism, it's a push mechanism.

When you take a new photo on any iCloud enabled device with Photostream turned on, it will inform whatever gubbins sits in the cloud which then accepts an upload of that photo, and pushes it out to other devices which are also registered.

As such it can never cater for pictures that were taken before the device was registered to use the service.  It has no way of performing the checking to see if you already have the photo, or options to perform what would effectively be 2 way sync at all, it has precisely no knowledge of your previous photos, and will merely start throwing down photos that are new after being registered to receive the updates.

So, in short, you can't.  To give you an analogy, it's like iMessages.  When you register a new device to receive messages, it picks up from that point onwards, and doesn't catch up on your previous history of messages.