Quick Notes

1. The bare infinitive is used as the main verb after the dummy auxiliary verb do, or most modal auxiliary verbs (such as will, can, or should..)

Examples:
I do know him (bare infinitive is the word know)
I do like you.
I can do it . (bare infinitive is do as it is used with modal verb can)

2. Several common verbs of perception, including see, watch, hear, feel, and sense take a direct object and a bare infinitive.

Examples:
I saw it happen
I watched it happen

3. The bare infinitive is also used with several common verbs of permission or causation, including make, bid, let, and have. 

Examples:
I made/let/had him do it. 
(However, make takes a to-infinitive in the passive voice.
I was made to do it.

4. The bare infinitive is also used after had better.
Example:
You had better leave now.

5. The verb help is followed by the bare infinitive. 
Example:
He helped them do it. ("He helped them to do" is also possible) 

6. With the word why. 
Example:
Why say it?

1. The full infinitive can function as a noun phrase. In this case it is used as follow:

as a subject.
Example:
To fail is human, to forgive is divine.

as an object.
Examples:
I intended to marry her. 
He wanted to know the whole truth.

2. It can also be used like an adjective or adverb.

Examples:
This is the game to watch. (to watch functions as an adjective, modifying the noun game) 
This is the problem to think about. (to think about functions as an adjective modifying the noun 'the problem') 
He went to his friend's house to study. (to study functions as an adverb answering the question why he went to his friend's house)
He is ready to go. (to go functions as an adverb, modifying the adjective 'ready'.)

3. It is used to mean "in order to" to express purpose.

Examples:
You need to exercise regularly to lose weight. (...in order to lose weight)
He works hard to earn a lot of money. (...in order to earn a lot of money)