What is the difference between I'll get it and I'll take it. and I'll do it. ?Feel free to just provide example sentences.

Depending on the context, they can mean pretty much the same thing.

—I'll get it.

"I'll get the phone." (answer the phone. This is actually a very common way to announce to anyone else present that you will answer the phone: "I'll get it.")
"I'll get the milk." (buy or pick up the milk)
"I'll get the delivery." (If you are talking to someone who needs several things to be done, you can say this as a way of taking responsibility for making the delivery. If you are talking to someone who is expecting to receive the delivery, this means you will go get it [from the front porch or from the mailbox or from the post office].)

—I'll take it.

"I'll take the call." (answer the phone)
"I'll take the delivery." (If you are talking to someone who needs someone to deliver an item, this means you will deliver it. If you are talking to someone who is delivering a package to your office, this means you will sign for the package.)

—I'll do it.
This means that a necessary task is clear and you will do whatever it is.
"I'll get it" can also be used in this fashion, it is casual.
Also: "I'll take care of it." This can be for a more formal setting.