I have seen on some airlines (mainly Emirates, but I think one more has it but the name escapes me) tag your bags with priority clearance if you are flying business or first.
However, it is not guaranteed to be the absolute first bag unloaded - you are just guaranteed it won't be dead last; and even then only at the airline's hub.
Bag load configuration depends heavily on the aircraft. Some larger aircraft carry standard size cargo bins which are loaded at random, and then distributed based on the current configuration of the aircraft (the exact load order is determined by the load master).
Upon arrival, one (or multiple) doors are opened and luggage is unloaded; typically from one side of the aircraft - it is almost always the opposite side from where passengers are disembarking.
If your flight is transit; then you might see two doors open where one is for loading and the other unloading luggage.
Luggage then makes it to the luggage loading bay, where it is put on a conveyor belt; and depending on the airport you might have multiple miles between the loading bay and the final deposit chute where luggage arrives.
During this travel time for luggage, it may be subject to multiple scans, stops, re-routes based on the particular airport - or, as is the case with some airports, it might just be some bloke chucking your bag through a hole onto a belt, which then meanders down to you (I'm looking at you, ISB).
Anyway, all that means is that there is no rhyme or reason for the order in which bags start arriving at the collection turnstile.
Some things that do make a difference:
Luggage checked "gate side" (usually, baby strollers and the carry-ons on full flights) are unloaded separately from the normal luggage, and this means they usually arrive quicker.
Soft bags (as others have mentioned) may be offloaded quicker - but this depends entirely on how they were loaded in the first place.
Odd-sized luggage or over-sized luggage is usually handled separately.