You are, of course, familiar with infinitives of purpose:
I went to the shops to get some bread.

They could also be used with in order or so as:
I went to the shops in order to get some bread.
I went to the shops so as to get some bread.

Note that negative infinitives of purpose cannot be used without them:
I went to the shops in order not to run out of bread.
I went to the shops so as not to run out of bread.

Infinitives of purpose work with nouns, too.
Have you got a key to open this door?
I’d like something to stop my toothache.
I need some more books to read.
Is there any milk to put on the cornflakes?
Did you tell her which bus to take?
Is there anything to drink?
I gave her a paper to read.
He needs a place to live in.

In some cases, what looks like an infinitive of purpose is, in fact, NOT one, but rather a way to indicate a sequence of actions.

Study these examples:
The patient woke up to see her sitting by the bed.
The workers went on strike to be arrested by the police.
They arrived at the port only to find that their ship had gone.

Infinitives can follow certain adjectives:

I’m pleased to see you.
John was surprised to get Ann’s letter.
She’s anxious to go home.
We’re happy to be here.
I was shocked to see how ill he was.
Most people are afraid to hear the truth about themselves.
We were right to start early.
Be careful not to wake the children.
I was stupid to believe him.
It’s very likely to rain.
She’s certain to win.
You were lucky not to be killed.
You’re welcome to stay as long as you like.
The letters are ready to sign.

The first / last / oldest / only / etc. can be followed by an infinitive instead of a relative clause:

He’s the oldest athlete ever to win an Olympic gold medal. = who won
Who was the first person to climb Everest without oxygen? = who climbed
The next to arrive was Mrs Patterson. = who arrived
She’s the only scientist to have won three Nobel prizes. = who has won

Infinitives are commonly used with either of the two structures:

Such structures often end with a preposition:

• She's nice to talk to.
• He's very easy to get on with.
• It's not a bad place to live in.

In certain cases, such infinitives do not carry any significant information:

• That's a good book to read. = That's a good book.

In case you forgot: infinitives are also used with too and enough:

• The apples were ripe enough to pick.
• The box was too heavy to lift.
• There was enough light to see what I was doing.
• There's too much snow (for us) to be able to drive.

Note that enough is often dropped before room and time:

• There's hardly (enough) room to breathe in here.
• Do you think we'll have (enough) time to do some shopping?

In some of the examples above, you were able to see for sb before the infinitive. That happens a lot when you need to indicate the logical subject of the action expressed by the infinitive:

• It's good for you to be home. = It's good that you are home.
• Japanese is difficult for Europeans to learn.
• Cricket is not very interesting for me to watch.
• The apples were ripe enough for us to pick.
• The box was too heavy for him to lift.
• He needs a place for his son to live in.

Infinitives come in handy in conditional sentences, making the possibility less likely:

• If he is to come, tell him I want to see him immediately. = if he comes
• If you were to give a talk in front of a large audience, what would it be about? = if you gave

There are also set phrases that include infinitives. Here are a few examples:

• Come to think of it. = It's just occurred to me.
• dressed to kill = dressed particularly smartly
• enough … to go round = enough for everyone present
• hate / dread / shudder to think
• have yet to do = haven't done yet but probably will
• months, years, etc to come = the following months etc
• needless to say
• not to mention
• nothing to be ashamed of
• nothing to write home about = nothing important
• remains to be seen = will be figured out in the future
• so to speak/say
• Strange to say = It's strange that
• to be honest / fair / exact
• to put it simply / mildly / bluntly / another way = to say it simply etc
• to say nothing of
• to say the least
• to start/begin with