How do I defeat shields?

Simply smash them with firepower. Bombers, Frigates, and Destroyers will provide plenty of support fire that can be beneficial to whittling them down whilst Artillery deals the final blow. Obviously, you'll want to smash the DeckShield building that puts them up to snuff them out for good. LasCannons are really good as they can do major damage to shields, but also lock down slots to prevent the enemy from putting new shields down as they'll get destroyed by the beam.