Which side of liquid gas interface is at high pressure? With respect to what (whether liquid or gas) should I consider that the convex side has higher pressure and concave at lower pressure? As seen in the example of image, it is given that air bubble inside water has higher pressure which implies that concave side is at higher pressure but in text,  it is quite opposite. What I am thinking wrong?

 A: I agree that it is confusing but I think that the author looked at the gas which is bulging out and called that surface the convex side (cf convex glass lens) and then looked at the liquid which is concave in shape.
A: You need to read about Laplace Pressure which is "the pressure difference between the inside and the outside of a curved surface that forms the boundary between a gas region and a liquid region". And given by

where $\gamma$ is surface tension. For a droplet of radius $R$ the above equation reduces to

The pressure difference between outside and inside is balanced by the surface tension of the interface. 
A: the pressure inside the bubble tends to increase surface area, the pressure outside tends to compress the bubble.
surface tension acts against expansion.
I.e. surface tension is compressive.
hence pressure inside has to be higher than outside.
the problem arises from defining "inside" and "outside".
for consistency "inside" a  surface is defined where the normals to the surface converge and "outside" is where they diverge.
