How do I properly place handicap stones?
The idea behind playing in the upper right first is so that White doesn't have to reach far to play his first move.  The corner immediately in front of him on his right is left open.

So the third stone goes in the lower right corner, from Black's viewpoint.

[Edit as per the comments: Move order for 9 stones]

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$$| . . . 4 . . . . . 7 . . . . . 1 . . . |
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$$| . . . 6 . . . . . 9 . . . . . 5 . . . |
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$$| . . . 2 . . . . . 8 . . . . . 3 . . . |
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