What wire type & size for sub-panel in a shed?
Well, I would not say you're doing anything stupid.  You have some very good questions.  

First,


If you did decide to direct bury the wire the minimum depth for direct burial is 24", not 18". 
At least three #6 ( black, red, white) and one #10 insulated ground ( green coating).
Anything &lt;= 60amps just requires a #10 insulated ground with green coating. 


Second, Consider voltage drop:


Load: 30 Amps @ 240V Single Phase.
Length of run: 80 feet
Wire Size: #6 Copper
Voltage: 240V
Voltage Drop: .81 %
Voltage At End of Circuit: 238.05


That is less than 3% which the NEC recommends for a feeder. Very Good !

Note: I would hesitate to install the ground rod if you have a in-ground pool in line of the transformer.

Correction: This being a feeder would require a grounding electrode!