We can get after something that needs tending or requires our action. That is, we see that the thing gets done. We tend to the matter.
I need to get after the backed-up laundry now that the washing machine has been repaired.
We can also get someone after something:
We need to get the dog-catcher after these packs of stray dogs.
And, in an extended, somewhat joking sense, we can get something after something:
I need to get the rake after those dead leaves.
That is, I need to rake those leaves. The rake, being an inanimate object, cannot be given an assignment or a task to complete as the dog-catcher can.
And we can get after something with something.
You need to get after those tall weeds with a sickle. A lawn mower won't do.