What is the difference between "revenge" and "vengeance"?

Is it okay to skip "with you" here to keep this sentence have the same meaning?

Yes, it would be okay to drop "with you" in this case and just say "take it on your morning jog" instead. It is okay to do this because the verb "take" in this case already implies that you are going to bring something with you, so you don't need to repeat this here. However, saying "with you" often adds emphasis to the fact that you are bringing something somewhere, and so it is often most common to say "take with you" instead of just "take". Here's an example of "take" and "take with you": Ex: Take these books for the trip. Ex: Take this book with you for the plane. Both sentences mean that someone is bringing something with them, but it is more emphasized with the phrase "take with you".

Why does the phrase "you got it bad" means "you're crazy about it"? I think the word "bad" sounds negative but the phrase "crazy about it" implies something positive.

Yeah, I see what you're saying! In this case "bad" doesn't mean something negative. It basically means that someone has strong feelings or emotions for someone else, usually associated with love or liking someone. A potential link is that if someone "has it bad" for someone else, it may not be good for them to feel those strong emotions- like it's obsessive. However, the function of "bad" in this case is to emphasis the amount and the intensity of the emotion, rather than the negative mood. I'm not 100 percent sure why it's used this way! But language develops and evolves as time goes on. So slang like this changes and develops too. Ex: Did you speak to Jane? She has it bad for Jonathan. => an intense crush or liking. Ex: I have it bad for Harry Styles. => To be obsessed with and to like a famous artist.

Why "agreed" was used here? Shouldn't it be "agree"?

The word "agreed" in this case is the adjective form of "agree". So "agreed" means that something was already accepted or decided. The adjective form "agreed" is therefore used to indicate a decision or an agreement that has been made. Ex: We're leaving town at 9 am tomorrow. Agreed? Ex: A: This weather is nice. B: Agreed!

What's "joke" mean here? I thought it has a positive connotation.

A joke on its own is a funny, good thing, because its meant to be humorous and entertaining. Being called a joke, however, is not a good thing. To be a joke means to be considered inadequate, worthy of ridicule, or a laughing stock. It's a good thing to tell a joke, but not to be one! You can also use "joke" to refer to things that are ridiculous or inadequate. Ex: This whole class is a joke. The teacher doesn't bother to help us when we have questions. Ex: I was upset because my friend called me a joke.

Does "at once" here mean something like "immediately"?

Yes, that's right! "At once" does mean "immediately" here. It's also similar to "right now." Ex: You must come at once. It's an emergency! Ex: She'll see at once that we're hiding something from here.