What insulation should I use when converting a metal shed to a pool house?
Cheapest solution (build cost) - don't insulate, costs a lot to run and performs poorly.
Cheapest solution (lifetime cost) - insulate well, costs more to build, less to run, performs better.
When to install - after your electrician and plumber are done, because doing it before they are done is going to increase your cost for them to do their parts of the job, and you'll have to repair the insulation after they are done.
Type of insulation - there are many. They all work, they all have costs which vary with your location and method of installation, they have varying levels of performance which sometimes but not always follow the varying levels of cost. There is no one right answer to be given here, it's a choice you have to make.