I unlocked bombing already but how do I buy it for my ships?

Once new weapons are unlocked with research each ship needs to be upgraded at either a shipyard or an arsenal.

Travel or send your fleet to a system with one of those stations. If the ship is part of you main fleet you need to dock it at the right station. Otherwise you can select upgrade from the fleet screen. Then select a mount point you want to change out and purchase the new weapon for it.