“I have to admit, I do prefer bottled water.”

You're talking with a friend who's very serious about doing things to improve the environment. You mostly agree with him, but he mentions that bottled water causes a lot of pollution. You always drink bottled water, so you say this.

I have to admit, I do prefer bottled water.

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I have to admit, (clause)

You start a sentence with "I have to admit..." when you're telling people something that you feel a little bit guilty about:

I have to admit, I'm pretty bad with remembering names.

I have to admit, I thought you were kind of annoying when I first met you.

You don't use this for things that you feel really guilty about, like making a big mistake at work.

I do (do something)

If you're just expressing your opinion in a normal sentence, you say:

I prefer bottled water.

I make a good living as a waiter.

You say "I do (do something)" when you're contrasting the things you do with the things that you don't do. So, for example, if someone said that you shouldn't use bottled water, this sentence:

I do prefer bottled water.

...is a contrast to what that person said. 

Likewise, this sentence:

I do make a good living as a waiter.

...can be used if someone says that waiters don't make a lot of money.

prefer (something)

To "prefer" something means to like it more than something else. This is used to compare two things. In the example at top, the speaker compares bottled water with tap water.

If you want to name both of the things you're compared, you say:

I prefer bottled water to tap water.

Or

I prefer bottled water over tap water.