You can't find Shaymin directly in Heart Gold, you'll have to trade your Shayim from Pearl/Diamond/Platinum onto your Heart Gold/Soul Silver.
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It needs to touch a Gracidea, which is a flower that can be bought from the Goldenrod Flower Shop. That is semi-correct. If you bring an event shaymin (not a hacked wild shaymin) to the Goldenrod Flower Shop as the lead Pokemon in your party the girl will give you the Gracidea.
There is no best Pokemon on Pokemon HeartGold.
Shaymin cannot evolve.
there is no Pokemon Mansion in Heartgold.
They are not in Pokemon HeartGold