The pedestal you install meets the definition of 'separate structure' and needs a grounding electrode.
The trailer is listed as a unit to be supplied cord and plug without a ground rod at the trailer.
The size wire you choose is very much up to you and how much voltage drop you want to live with.
You mentioned the trailer is set up with a 30 amp plug, I will guess the load will not be more than 22 amps and that you could live with 5% voltage drop.
Using those figures you would need to run 3 AWG copper.
If you ran 10 awg for this distance and 22 amps of load you will have about 20% or 24 volts of loss.
To make a good decision on the wire size you must know what the connected load will really be.
