“I'll get to that in just a minute.”
You're giving a presentation at work. One of your employees asks a question. You have slide in your presentation which answers that question, so you ask her to wait.
I'll get to that in just a minute.
When someone says "a minute", they usually don't mean exactly 60 seconds. They mean a short length of time, somewhere between 30 seconds and 5-10 minutes.
If you want to specify exactly one minute, use the phrase "one minute". The sentence:
Can you give me just one minute to finish writing this?
...sounds much more specific about how long you are going to spend writing the message.
It's common to add "just" to this phrase:
A: Are you coming?
B: Yeah, just a minute!
Use "that" to refer to something that a person just said:
A: I'm going to try to start a career as an actor.
B: So does that mean you're moving to L.A.?
Or you can talk about something that you heard on T.V. or radio using "that".
When you use "get to ___", it sounds like you have a long list of things to do, and you're finally reaching one item on the list. You use this phrase when you're busy:
I'll try to get to that this afternoon.
You can also use this phrase when you have a lot of things that you want to talk about in a conversation, a speech, a meeting, etc.
I wanted to talk about the new marketing campaign we just rolled out, but it looks like we won't be able to get to that today.