I notice that some PDF's I receive I am able to edit the text that is already in the document while using Acrobat Pro. However when I save my own PDF files I cannot seem to edit the text. Is there a specific way I should be saving my PDF? Either on Photoshop or inDesign?
Photoshop - no chance.
InDesign - will usually just work.
Text editing in Acrobat is, however, a desperate last resort. Really, never, never consider it for files where the original is still available, even at some effort. Text editing is for where the original is lost and all the backups eaten by termites.
OR where its minor touchups. Such as a letter left out of or off a word. Or one or two words left out. Or a Duplicate word or two.