The infinitive

Infinitives are the basic form of the verb and often follow "to". Infinitives with the word "to" are called full infinitives and can be the subject of a verb. To read a good book is a great way to relax. They can also be the object of a verb. I don't want to go.

We use the infinitive after certain verbs.
We can't afford to go abroad. We agreed to help them. I decided to do it myself. I asked to see her boss. She expects to make a lot of money. We hope to start next week. Would you like to go abroad this year? I need to make a new plan. I promise to tidy up later. We had to work really hard to pass the exam. I want to find a better job. He refuses to help her.

We use the infinitive after certain verbs plus a noun/pronoun
I advise you to leave. He expects you to work harder. I want you to wash the car. I invited John to come for dinner. I need you to help me move the bed. He told me to see a doctor.

We use the infinitive after certain adjectives.
I was afraid/anxious/prepared/ready/willing to go.
I was amazed/astonished/determined/glad/ happy/lucky/pleased/surprised/ relieved/sad/shocked/sorry to see him

Bare infinitives are infinitives without "to".
They are used after modal verbs such as can, could, will, would, may, might, must etc. She can write very well.
They are used after would rather and had better. I would rather stay at home. He had better leave. They are used after let and make. Let me explain what happened. Make him leave.

Gerund

A gerund (often known as an -"ing" word) is formed from a verb but functions as a noun.
Gerunds can be the subject of the verb. Playing bridge is a lot of fun.
Gerunds can be the object of the verb. I like playing bridge.
Gerunds can be the object of a preposition. I am looking forward to playing bridge this evening.

Here are some verbs plus a preposition followed by a gerund:
to insist on, to apologize for, to think of, to take part in, to object to

Here are some verbs plus noun/ pronoun plus preposition followed by a gerund:
to have difficulty in, to prevent him from, to stop someone from, to thank someone for, to blame someone for, to accuse someone of

Here are some combinations of the verb "to be" plus adjective plus preposition followed by a gerund:
to be interested in, to be excited about, to be used to, to be tired of, to be good at

Here are some examples of verbs that can be followed by a gerund:
I admit spending all my money on rubbish. I avoid seeing him as much as I can. I considered going on a diet but decided against it. I deny saying that. I enjoy being with him. When I have finished reading this through, I'll give you a hand. I often go dancing. I don't mind working at the weekend. I hope they stop making all that noise, I can't hear the telly.

Both gerunds and infinitives can be the subject of a sentence. Writing/To write English is difficult. Both gerunds and infinitives can be the object of the verb. I like speaking/to speak English. But only gerunds can be the object of a preposition. She is thinking about going to France for her holiday.