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A liquid drop near the surface of a solid experiences a force from the liquid interior molecules as well as due to solid molecules near the liquid-solid interface. Forces due to gravity and air are neglected.

Figure (a)

In figure (a), the two vertical lines denote a solid and the right part to it is the liquid. The adhesive force between the solid and the liquid is greater than the cohesive force. $F_\mathrm{S}$ is the force due to the attraction of solid molecules. $F_\mathrm{I}$ is the force by the liquid molecules. $W$ is the weight of the small part of the liquid considered. My book says that the resultant force $F$ is as shown and the liquid surface should be perpendicular to it.

But I think that if the liquid has to rise upwards, there must a component of force in the upward direction initially to pull the liquid upwards and I don't see any.

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The diagram that you have provided can be used to measure the angle of contact of the fluid with the interface but you are trying to relate it with a phenomenon that is caused primarily due to surface tension which is a different concept.

The liquid rises in the capillary due to surface tension, a force that comes into play owing to the gradient of molecules at the interface (for more information, see Why is surface tension parallel to the interface?)

This force causes the liquid to risesurface tension

Here T is the force by the liquid on the tube due to surface tension and R is the equal and opposite force by the tube onthe liquid, pulling it up.

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