on (one's) way home

"On my way home" refers to anything that you do while you're coming home from somewhere, like from work, from a vacation, from shopping, etc.

The phrase "on my way ___" is usually followed by "to ___":

I was on my way to work when she called.

But "home" is special because you never use "to" with it:

I'm going to Ted's house.

I'm going home.

"Here" and "there" are also special in the same way:

He's on his way here right now.

pick up (something)

The phrase "pick up ___" can mean to buy something at a store. When you use this phrase, it sounds like you are buying something quickly while you are on your way to somewhere.

Hey

You say "Hey" at the beginning of a sentence in casual English when you want to get someone's attention. For example:

Hey, Hitomi, can you hand me that box over there?

It's common to say "Hey" before you ask someone a question, or when someone has done something that's wrong or unfair:

Hey, you're taking up the whole bed!

Use "hey" with people you know or are familiar with. When you're talking to people you don't know as well, "Excuse me" is more polite.

take-out

"Take-out" is food that you buy from a restaurant, but instead of eating it at the restaurant, you take it home to eat.

In the U.S., the most famous food for take-out is Chinese food. Mexican food, Indian food, and pizza are also popular take-out foods.

Use "take-out" like this:

Why don't we get some take-out?

I don't cook very much. I usually just get take-out or snack on whatever I have in the fridge.

You can also spell this word as "takeout".