Reaction of soap with water
The simplest kinds of soap are just a fatty acid chain with something more ionic on the end - something like the ubiquitous sodium laureth sulphate: 



When it's in bar form, it behaves a bit like a solid block of plastic, because chemically speaking, those long chains of carbon are quite a lot like a common plastic such as polythene. These chains attract each other and repel and exclude water, keeping the block together. The sulphate group on the end is hydrophilic, though - it's chemically stable surrounded by water. This means that when you add water, it infiltrates and teases apart the structure of strands, getting to the ionic ends on each chain and forming a "solvation shell" of waters around each one. There's no real chemical reaction here - as you say, it's a physical process, which is why rubbing the soap helps - you're applying shear to the structure and helping to separate the strands.

Once they're separated from the bulk, the soap forms tiny bubbles called "micelles", which have the chains pointed inwards towards each other and the ionic sulphate groups meeting the water at the outside.