From pollen grain to ovule

Pollination ended with a pollen grain sitting on a stigma. The male gamete inside it is still a long way from the ovule, and it has no way of moving. The plant solves this by growing a tube.

In pollination, the pollen of a flower is deposited on the stigma of that same flower or of another flower of the same species.
After that pollen is deposited on the stigma, it is stimulated by the sugar solution present on the stigma, and pollen germination begins.
In pollen germination, the pollen tube grows down along the style and reaches the ovule within the ovary.

Why the stigma is sticky and sweet

The stickiness holds the pollen grain where it lands. The sugar does two jobs at once: it is the signal that starts germination, and it is the food that pays for growing the tube. A grain that lands anywhere else gets neither, and nothing happens.

The gamete never swims

Unlike an animal's sperm, a plant's male gamete has no means of moving itself. It is carried down inside the growing pollen tube. The route it takes is exactly the three parts of the gynoecium in order: stigma, style, ovary.