Tuff Shed and adding climate control
You probably don't need to worry about a vapor barrier in your climate.  Most building scientists recommend against them since they impede drying of the structure in at least one season of the year. (They are still required for northern climate zones)  While it would be best to have a continuous layer of insulation under the siding, homes have been built without such for many years. Between 40-50% of your energy escapes through air leaks, not insulation and while air sealing the sheathing is easiest it is still possible to do a good job from the inside.  

The best solution would be to spray foam the wall cavities, ideally with a closed cell foam.  This gives you a nice airtight seal, and the foam has a high R value per inch.  This is, however, probably cost prohibitive for a small project. (If you could combine it with upgrading your home insulation it might be worth considering) 

If you are handy I would recommend "poor man's spray foam".  Basically you fill the walls with sheet foam panels that are cut slightly smaller than the width between studs.  The gaps are then filled with canned foam to make a nice airtight seal.  This gives much of the same benefits as spray foam but it is slightly cheaper and more DIY friendly.  

Once you have the walls insulated you can put up a drywall or interior paneling, again working to make the system as airtight as possible.  

If the building is nice and air tight you should be able to climate control with a mini-split heatpump and you will have a very comfortable space.