Теория:

Study the information about verb + -ing or to. Part 1
 
1. Some verbs can be followed by to and -ing with a difference of meaning.

Remember:

I remember doing something = I did it and now I remember this. You remember doing something after you have done it.

I know I locked the door. I clearly remember locking it.
  
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I remembered to do something = I remembered that I had to do it, so I did it. You remember to do something before you do it.

I remembered to lock the door, but I forgot to shut the windows.

Regret:
 
I regret doing something = I did it and now I am sorry about it.

She regrets saying what she said.

I regret to say/to tell/to inform you = I am sorry that I have to say.

We regret to inform you that we cannot offer you the job.
  
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Go on:

Go on doing something = continue with the same thing:

We need to change. We can’t go on living like this.

Go on to do something = do or say something new:

After discussing the problem, he went on to talk about the project.

2. Use can use -ing and to after these verbs:

Begin, start, continue, bother, intend.
  
It has started raining  or  It has started to rain.
  
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Источники:
www.newdoorideas.com
simply-360.co.uk
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