There are a couple of standard conventions about commas that you need to know, and a couple of optional, judgement calls.
You have two simple sentences that you are connecting with the simple conjunction and. (Your sentences are commands, but commands are sentences.)
Please verify the changes, and verify the whole application.
This really is just commas at the most basic level: There is a comma before and—always; for everyone, this is the rule.
Some people don't put commas before coordinating conjunctions, but that is not your issue.
Of course, if you write something that is not a compound sentence; you don't need a comma
Please verify the changes and the whole application.
Your second issue, which you also have to handle, is how to insert the also.
If you had a longer insert, it would need commas around it. For example
Please verify the changes, and, while you are at it, verify the whole application.
With short insertions, this isn't mandatory, and it's usually too many commas. You can think of it as an adverb. So, you have a choice; it's either
Please verify the changes, and, also, verify the whole application. (I think too many commas) or
Please verify the changes, and also verify the whole application.
One more choice for you to make: There is a 'rule' that you can add commas to indicate how it is spoken or to make it easier for the reader to understand.
Please verify the changes, and also, verify the whole application.
If this sounds or reads better for you, use it. It is my choice, but you are not me.
The goal is for commas to be your friends.