In Boom Beach you can only attack islands visible on your map. There is no "search for opponent" function. Player opponents, referred to as
Blackguard mercenaries in the game, are found when exploring the map (and as invasions on your base).
Matches are found by comparing your, and your opponents
Victory Points. This should match you up evenly with other Commanders in the archipelago. Victory points are gained by defeating other bases in battle.
Upon freeing islands, the local natives will send you gold to show their gratitude. Whenever you free an island, be aware that the evil forces will try retaking it, so you might have to send in your troops again. The Blackguard is persistent after all (and stylish, I might add).
Once you have freed all the islands on your map, you can either
wait for the Blackguard to strike back, or upgrade your radar to
explore more of the archipelago.
Attacking will cost you gold, the cost increasing whenever you level up.