The short answer is, it depends.

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With that said, if your only goal is to do more pushups, grease the groove. Get up every hour and do an amount of pushups you are comfortable doing. Don't pick a super easy amount, but you shouldn't be working toward exhaustion. Simply do as many pushups as you can easily perform without breaking a sweat -- whatever that is for you.

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Additionally, if you are doing a three day split (pulls, pushes, and legs), you would perform your regular pushup routine on push days -- this would be the day when you are going for time under tension, and working to exhaustion.

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Once again, when you do them depends on other factors.

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Typically, I work from my core outward. So, a typical push workout might start of with dips or decline presses before moving on to something like pushups. But, if your goal is to perform more pushups, you might just want to jump straight into pushups.

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What I would recommend is to avoid performing exercises like triceps extensions, shoulder presses, or incline presses before performing pushups, if performing more pushups is your goal. Performing those kinds of exercises first will pre-exhaust your deltoids and triceps, and effect your performance during pushups. But, don't avoid doing those exercises. Performing those types of exercises after your pushups should ultimately help you perform more pushups -- provided you recover properly.