I don't think you will be able to harden the mild steel shell with an oxy-acet torch. At least not to any degree that will make much difference. I assume you want to make a hardened shell to make it more difficult to get into the safe. I think you will need to use a hardenable alloy which will need to be heat treated in an oven in order to control the process. Along with a high alloy material comes potential welding problems.
I think your money would be better spent on a commercially available safe rather than on special alloys, welding, and heat treating. If you just want a box that is tough to get into under normal circumstances, then I would use the 3/16" mild steel as it is and do what you can to protect the hinges and lock mechanism.
A thief can find ways to cut open most any safe given time and equipment if he thinks it is worth the trouble.
CHGuilford

zeN
There are lots of ways to harden mild plate with a torch. The simplest is to adjust your torch to a carburizing flame(red orange flame tips) and then start heating the plate evenly one full side at a time, moving in from the outer edges. The most rapid carbon absorption occurrs at bright cherry red but it will occurr at lower temperatures albeit MUCH slower. This technique is called flame hardening and was once commonly used and is still used today on things like very thick shafts where it is impractical to put the entire object in a furnace with a carbon rich atmosphere. The only drawback in your application will be the possibility of plate distortion unless it is a very small safe. Another option is to use the same basic flame technique and a commercial case hardening compound like Kasenite. You would heat the area and then sprinkle the compound on to the plate where it would stick like a flux then you continue to apply heat for the specified time. Be cautious with these compounds as they are usually pottasium cyanide based. Whatever method you use to increase the carbon in the surface you then need to quench the object rapidly from cherry red to harden it. What you end up with will not easily be cut by a HSS drill bit. What you are propossing is a lot of work but if your just having some fun don't be dismayed by naysayers and new tech. There were cultures case hardening steels 2000 years ago in nothing more exotic than a pit full of wood coals.
Hope this is of some help
Sean