Open World p 20. Gerunds and Infinitives. Grammar

The infinitive with to is used:

• to express purpose: E.g. I went out to get some fresh air.
• after some adjectives: E.g. It's not easy to find work these days. I was happy to help.
• after the verb to be, to give orders or to express an arrangement: E.g. You are to stay here until I get back. The President is to visit Poland next week.
• after would hate/like/love/prefer, with or without an object: E.g. Would you like me to do it now?
• after the following verbs: agree, appear, arrange, ask, attempt, choose, decide, demand, deserve, expect, fail, help, hesitate, hope, learn, manage, need, offer, prepare, pretend, promise, refuse, seem, threaten, want, wish. E.g. If you need any help, don't hesitate to contact me. They hoped to negotiate a better deal.
• after the following verbs + object: advise, allow, ask, cause, challenge, enable, encourage, expect, forbid, force, get, help, intend, invite, order, persuade, recommend, remind, teach, tell, urge (advise, persuade), warn. E.g. My family encouraged me to go to university. The police asked everyone to remain calm. If advise and recommend are used without an object, the gerund is used. E.g. I recommend her to apply to King's College. I recommend applying to King's College.
• after the following nouns: ability, attempt, capacity, chance, decision, desire, determination, effort, failure, intention, need, opportunity, permission, plan, proposal, refusal, right, tendency, it's time, way, willingness. E.g. It was the director's refusal to accept his proposal that led to his decision to resign.

The bare infinitive (without to) is used:

• after modal verbs: E.g. I shouldn't eat this really, but I can't resist it.
• after the following verbs: help, had better, let, make, would rather/sooner. E.g. We'd better go home now- it's very late. Can you help me tidy up, please? In the passive, make is followed by the infinitive with to. E.g. We were made to do all the dirty jobs.
• The verbs feel, hear, see, notice, overhear and watch can be followed by an object + the bare infinitive when we are describing a single action or the complete action. E.g. We saw a young mother slap her child in the supermarket ( = she slapped once) I saw Martha cross the road (= I saw all of it, the whole action from start to finish). These verbs are followed by an object + -ing when the action is repeated or you only witness part of the action. E.g. I saw a young mother slapping her child ( = she slapped him several times) I saw Martha crossing the road (I saw part of it). 

The gerund is used:

• as the subject, object or complement of a clause or sentence: E.g. Playing golf helps me relax but I find watching it on television rather boring. Smoking is bad for you.
• after prepositions. E.g. We thought about going to France this year. Are you still interested in buying the property?
• after the following expressions: have difficulty (in)/problems/trouble, there's no/little point (in), there's no use, it's no good/use, it's (not) worth. E.g. It's no use asking him- he won't know the answer.
• after the following verbs: admit, adore, advise, appreciate, anticipate, avoid, can't help, can't stand, consider, delay, deny, detest, dislike, dread (be afraid of), enjoy, feel like, give up, imagine, involve, keep, (don't) mind, miss, postpone, put off, practise, prevent, propose, recommend, resent (feel bitter or angry), risk, suggest. E.g. I resent /rɪˈzent/ having to do all the housework myself. Nothing would prevent him/his speaking out against injustice.
  appreciate doing something I don't appreciate being treated like a second-class citizen. 
  
• after the following verbs+ the preposition to: adapt/adjust to, admit to, confess to, get round to, get used to, look forward to, object to. E.g. She confessed to being surprised by her success.
•  Where the subject of the main verb and the gerund are different, an object (pronoun) or possessive adjective is used. E.g. appreciate somebody doing something We would appreciate you/your letting us know of any problems. I really appreciate you/your helping me like this. We appreciate you/ your coming to tell us so quickly. Dislike somebody/something doing something He disliked her staying away from home. Resent somebody doing something She resented him/his making all the decisions. Imagine somebody/something doing something I can just imagine him/his saying that! I couldn't imagine him/his eating something like this.

The following verbs can be followed by the gerund or the infinitive with to with no change in meaning: begin, can't bear, continue, intend, start, hate, like, love, prefer. E.g. She suddenly started singing/to sing. The infinitive is common for specific situations: I hate to say this, but your breath smells. I hate getting up early every morning.

The following verbs can be followed by the gerund or the infinitive with to, but with a change in meaning:

• The infinitive is used with remember, forget, regret, go on and stop when the act of remembering, etc comes first. The gerund is used when it comes second. E.g. I must remember to post this letter later. I distinctly remember posting the letter yesterday. Her car broke down and no one stopped to help her. I've stopped eating chocolate. When regret is followed by the infinitive with to, it is normally used with verbs such as say, tell and inform. This use is formal. E.g. We regret to inform you that your application has been unsuccessful.
  When followed by gerund, go on means to continue with the same activity. With an infinitive, it means to change to a different activity. Go on doing something He went on working well after normal retirement age. Go on to do something: to do something after completing something else. E.g. The book goes on to describe his experiences in the army. After her early teaching career she went on to become a doctor. After outlining the problems, she went on to offer some solutions.
• Try+infinitive with to means attempt. Try+gerund means experiment with. E.g. Please be quiet- I'm trying to sleep. If you can't sleep, try using earplugs.
• mean+ infinitive with to means intend. Mean + gerund means involve. E.g. I've been meaning to write to you for ages. Changing jobs also meant changing house. 

Passive infinitive or -ing form are used to describe actions which are done to the subject. E.g. I hate being told what to do.  It is expected to be finished soon.

Perfect infinitive or ing form are used to emphasise when one action happened before another. E.g. She mentioned having seen him leave. They seem to have solved the problem.

Negative infinitive or -ing form are formed with not. E.g. It's quite common not to understand at first. Not understanding is quite common. 

Related links:
Verb patterns list
Verb patterns Cambridge
Gerunds and Infinitives
Perfect English Grammar