Will doing pushups and pull ups develop strong arms?Yes, and much, much more!
Push ups and pull ups as a pair constitute a complete upper body workout. They build strength in your shoulders, chest, upper and mid back, elbow flexors and extensors, and your forearms.
Warning! Major plugs incoming!
One of the best answers on Quora of how to progress to a full push up (if you aren't there yet), is the following answer:
Sed Chapman's answer to How can I learn how to do press-ups gradually?
(I have Sed Chapman's permission to use that answer)
From full push ups, you can either go vertical
Handstand Pushups by Bhavesh Aggarwal on The Quora Gym v.P.
Bhavesh Aggarwal's answer to How many push-ups should I be able to do before moving on to one-armed push-ups?
Keep in mind that you can combine both things once you're strong enough, i.e. doing one armed push ups with your feet on a couch. You'll be quite a beast by then.
Coming to the pull ups side of the equation, here is a progression to your first pull up:
Bhavesh Aggarwal's answer to How do I increase my level of pull ups step by step?
From your first pull up, you can progress to more with this method:
What's the best way to get to 10 pull-ups if you can only do 1 right now?
And if you hit a plateau, this advice will help:
My pulls have been stagnant for a while, how can I break the plateau?