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!
If you do both of these exercises regularly then I assume you are quite strong at each of them. If that is the case, I’d say it doesn’t matter for you which exercise you choose to do first. If I was working with a new client or someone who was not as strong in these moves as you are, then I would have her do the pushup first and then the plank. Here is why. To keep it simple, the pushup uses chest and shoulders and also the abs in a similar way the plank trains abs.. So if I have a client do two sets of pushups for a number of reps that she finds challenging, I will have trained her abs in the process. Next I have her perform the plank and she finds her abs are now too tired to plank with good form. OK she is done with abs for today.. We have worked them to fatigue. As her strength improves she will eventually be able to hold the plank after completing pushups. However, if I have her do the plank first to fatigue the abs when she goes to try pushups she will not be able to stabilize her core well enough to perform pushups with good form. The result is her abs have been worked but her chest and shoulders have been cheated. Strength in the chest and shoulders will lag behind until the abs are strong enough to do both exercises well.
This is difficult to answer given the information, and so I would just say this:
You will be able to perform the first exercise at a higher intensity.
The second exercises uses many of the same muscles (not necessarily as prime movers, but as stabilizers).
Yes, absolutely you should do pull ups (all else being equal); any large muscle group exercise, such as squats, deadlifts, pull ups are going to be good ‘general’ options.
You say you are doing the exercises everyday. If you are doing the exercises correctly and to a fair degree of fatigue, you should take a day in between to recover properly. The planks use high endurance stabilizer muscles and once well conditioned, these muscle could be trained more aggresively. But you should have a specific need to do so. That is, you are having issues with stabilizing your spine due to low endurance capability of these muscles. Then you may want to lower the volume (the total time spent doing planks) in one session and add more sesssions throughout the day. Starting with one 1 minute session with 3 to 5 plank holds. And then increase the time held for each one by 1 or 2 seconds, building from there. Even if you are already doing more than this, it is not a bad idea to restart the program and make sure you are doing each exercise correctly.
Since I can’t see you personally, a lot of this advice would include variations specifically to what you are doing and how. And there are many muscle groups that you apparently are not exercising yet. Adding pull ups is possible, but do so cautiously. Pull ups are very difficult to perform correrctly if you are not already strong enough to do them. Again, adding 1 to 3 pull ups every other day for a while before attempting more. And adding one pull up at a time for a few sessions is wiser than trying to do as many as you can right away. Working too hard too soon is a recipe for injury. Give your body some time to adjust and prepare for more intensity and volume.
I highly recommend you let your doctor know you are interested in exercising more and more intensely. It can’t hurt and it could save you a lot of hurt down the road.