Order a takeout

“It's so much easier to order something than to spend all that time going to the grocery store, cooking, washing dishes... Not to mention I'm not that good of a cook to begin with.” 

You don't cook very much. Instead, you like to order takeout from restaurants. Now you're talking with a friend about why you do that.

 order (food)

You use the word "order" for when you ask for food at a restaurant:
What are you going to order?
You can also order things to be delivered to you, like food:
Why don't we order a pizza?
Things that get delivered in the mail can also be "ordered".

 not to mention (something)

Use "not to mention ___" when you're listing reasons or excuses. It marks things that are kind of "extra" items on the list.
For example, in the sentence at top the speaker gave two good reasons why smoking is a bad habit: it's bad for your health and expensive. The bad odor that cigarette smoke leaves is another reason, but it's not as strong of a reason as the first two. So the speaker adds that to the list with "not to mention".
The costs, not to mention the risks, of transporting the telescope to and from space would have been excessive.
You can follow "not to mention" with a noun like in the examples above. You can also follow it with a verb:
She'd never even held a gun before, much less shot one.
Or with a clause:
It's so much easier to order something than to spend all that time going to the grocery store, cooking, washing dishes. Not to mention I'm not that good of a cook to begin with.

not that (adjective) of a (something)

Use this construction when you want to:

• describe what something is not ("long")
• tell what category it's in ("a hike")

For example:
She's not that good of a singer.
This isn't that old of a building.
It really isn't that big of a deal.
Of course, you could also say:
It's not a very long hike.
She's not a very good singer.
But these versions put more emphasis on the description ("long", "good"), while the examples above put a little more emphasis on the category.

spend (time) (doing something)

To "spend time ___ing" means to use your time to do it. You use the word "spend" with time because we think of time as being very valuable, just like money.

go to the grocery store

The grocery store is a market or supermarket where you can buy food and other common household products. Use the phrase "go to the grocery store" to describe someone making a trip to buy things there.

(not something) to begin with

"Not ___ to begin with" means something like "not even ___".
Imagine that you were supposed to have the day off from work, but your boss asked you to come in at the last minute. While you're at work, a coworker asks you to cover her shift and work later. You might say:
No way! I'm not even supposed to be here today to begin with.