How do I determine how large of a filter to purchase?

I have a 24 foot above ground round pool and we found out the filter and other parts are cracked and corroded so we're going to just buy a new pump and cartridge filter.

How do I determine the best size filter and HP pump for my pool?

Are you sure you want a cartridge filter? That would be my last choice. A sand filter is by far the easiest.

I am not the pump/filter expert but I'll tell what I have. I used to have a 24ft AG pool and now have a 27ft AG. For both pools I have used the same pump. When it was time to replace my pump a few years back, I had one of the other moderators (Poconos) make a suggestion for me. He suggested I get the Hayward Superpump -- is a 1/2 hp full-rated inground pump. It is paired with a 300 lb. sand filter. Inground pumps are usually built a little more sturdily and it is also self-priming which I think most AG pumps are not. That is a big plus.

Someone else may give you a better suggestion. But, just remember, in the case of pool pumps, bigger is NOT better. You want a smaller pump paired with a bigger filter for best filtration.

Welcome to the Pool Forum!

Here's a system that would perform well for you, and be very economical:
Hayward S230T932S 23-Inch Pro Series Sand Filter System with 1-1/2 HP 2 Speed Power-Flo Matrix Pump

You'd want to run it on low speed all the time, unless you actually needed high speed. You would have to backwash on high, but might need to put a valve on the backwash discharge if the flow rate is too high, and is 'blowing sand' out.

One like this would work, though there are cheaper options, depending on your skills:
Hayward Trimline 2-Way Ball Valve, 1-1/2-Inch SKT Pipe PVC Material

There are also DE filter options. DE will produce clearer water, but is unforgiving if you have an algae problem: live algae 'plugs' DE filters almost instantly. You have to kill the algae before you can operate the filter.