In the rules that is covered under the several sections on Amphibious Assaults.
With the exception of Battleships (and Destroyers if you have the Consolidated Bombardment technology), NO naval units can participate in land combat.
So, assuming no technology, in your example:
The fleet sails up to the island.
The Battleship fires a single salvo onto the island while the TRN’s offload to the beaches
Attacker first combat roll is the BB and the offloading land units.
Defender fires for their 2 INF.
Second and subsequent rounds of combat are your offloaded land units against enemy units with no further BB shots.
If you had a Carrier along with FIGs on it, the air units could also participate in the battle.
If you did have the Consolidated Bombardment technology, both your Battleship AND the Destroyer would fire an openning salvo and then would be removed from the battle.
See page 15 of the OM for Battleship openning fire info
Pages 19-20 for Amphbious Combat
Page 9 for Combined Bombardment
Hope this helps :-)