How can I connect my new light fixtures to old fluorescent fixture bases?
Cut off the light bulb adapter, strip the wires, cut off (or release, if they are releasable - look for a hole to poke something into) the plastic connector blocks, strip those wires if you had to cut them off,  and wirenut black to black and white to white. Keep the little orange connector, it's handy for removing the light later, and for being able to wire the connections without the light being in your face.