How do I UV unwrap and texture onion-like object? What would be the best method to UV unwrap and texture an onion-like object? 

It doesn't have to be an onion specifically, it's just an example. I'm talking about all the objects that are spherical in shape and have distinctive parallel lines following the surface from one pole to the opposite pole, converging around both poles (like watemelons etc.).

I'm mostly interested with image-based texture solutions, not procedural ones.

The photo is Creative Commons licensed image taken by flickr.com/photos/user-colin/
You want to UV unwrap it such that you get a square/rectangular UV island. Then you can use an image texture that is setup for polar coordinates. The trick with a sphere is that you have to remove the single vertex at each pole to unrwap it properly. 


Create sphere.
Delete vertices at each pole.
Set viewport to ortho mode and view the sphere directly (front or side, no angle).
Unwrap via sphere projection. Your UV island should be square.
Create a polar coordinate image texture (features run from top to bottom).
Apply to texture using UV coordinates. I've used a Texture Mapping node to increase the X scale so that it tiles more horizontally.