Something is definitely drawing a lot of current, causing the fuse to blow out. Does it blow when you switch to defrost?
Could be a number of things - first thing I'd check is the refrigerant level in the AC system, make sure it is OK. If the level is too high, the compressor has to work a lot harder, causing it to put a lot of load on that clutch. If the clutch burns up - when it engages, it could draw a ton of current. Too low - the compressor shouldn't even come on, as a safety feature. More than likely - it is a faulty magnetic clutch that is causing the heavy current draw when you first power it up. You'll have to take it to a qualified HVAC place to double check it top to bottom - could be something really simple and not a bad clutch.
Could also be wiring that got shorted out somewhere - maybe the insulation is worn off at a point close to a chassis ground. Those can be a nightmare to trackdown - as you have to trace back each wire by hand to make sure it is not shorted.
Anything else that happened to the car recently? New audio system, any recent electrical work, did it do this before, did it pop fuse for any other reason/different component?