Push ups and planks are both great exercise for your core but target two different things.
Push ups are done primarily to increase muscular endurance, strength, or hypertrophy (of the chest, shoulders, and triceps). A plank is an exercise done primarily to increase core stability.
The order depends on your goal, but personally I would complete the planks first and then push ups. I would consider the plank as a core activation exercise and a warm up exercise to the push up.
I do recommend adding some type of pull exercise (pull ups, rows, or any other exercise that target the back).
I would recommend performing an exercise for each movement of the body… Push, pull, squat, hinge, single leg, loaded carry, rotation, and core stability.
Hope this helps!