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I can know some patterns like these.
I enjoy cooking.
He wants to swim.
But for other verbs like admit, allow, agree, appear and so on. It seems that I have to remember which verbs are followed by infinitives or gerund.
I know I have to listen a lot. At least I know how to use "can't help" from this song "but I can't help falling in love with you...."
But for learner do we have only to remember them or are there any other solutions?
I will try to answer this from a learner's point of view.
I agree with @StoneyB. In the end, you will need to know each and every one of them. Up to this point, I couldn't find "one rule to cure them all" yet. But after spending years learning English, at some point I realized this very useful guideline on my own. This guideline helps me choose the correct pattern most of the time!
Here is my guideline,
You need to understand the subtle difference between "I stop thinking" and "I stop to think", and you should understand it really really well.
If you understand that well, you will be able to figure out why "he was to leave" means the same to "he was going to leave". For example, Clark announced that he was to leave the show to concentrate on his music career. You can also rephrase it to ... he was leaving the show..., and it will still have roughly the same meaning. Just like your case of "like cooking" vs. "like to cook".
Although "I like cooking" and "I like to cook" would mean the same to most native speakers, you can think of "cooking" and "to cook" differently. And you can still be right!
Personally, when I heard "cooking", I will think of the act of cooking. When I heard "to cook", I will in my mind see myself (or someone else, depending on the context) getting up and start to cook. Either way, I will end up cooking, which is why I believe thinking this way is quite close to native speakers.
That's the way I see it.
The most common use of infinitives is, as you've recognized from the pattern in your examples, when they follow another verb. Most of the time, when this happens--if the verb is transitive--the infinitive is the either the direct object or it's part of a phrase that is the direct object. (e.g., I [have/had] agreed to ride with her I agreed to do what? To ride with her).
And while they're technically verbs, infinitives can function as nouns/adjective, just as gerunds can. So ask yourself, What part of speech is this?
e.g., To be prepared first is a good idea [subject is to be prepared; direct object is a good idea]
If it's a complement, an object, or the subject, infinitives can be used.
Also, I use bare infinitives frequently in the subjunctive mood, which is another very common use.
e.g., I suggest you be home when I get there. Were I you, I'd be there.
Additionally:
It seems that I have to remember which verbs are followed by infinitives or gerunds.
Yes, that's correct, although I wouldn't necessarily call them gerunds (because they're not used as nouns/adjectives). Words with the -ing suffix function as verbs when they immediately follow an auxiliary. They're used to form the continuous/progressive tenses.
But for other verbs like admit, allow, agree, appear and so on. It seems that I have to remember which verbs are followed by infinitives or gerunds.
It's still the same rule. The infinitive functions as a different part of speech.
e.g.:
I am trying to agree with you. [present progressive; direct object is to agree]
You appear to be angry. [object is to be angry]
She has been allowing him to sleep on her couch. [present perfect progressive; object is to sleep (on her couch)]
I was planning to go out. [past progressive; to go out is object]
They were walking to lose weight. [to lose weight is the object]
I have been drinking to get drunk. [present perfect progressive; to get drunk is the object]
Before I began to careen down the mountainside, I had been trying to ski. [to careen down the mountainside is the object of the dependent clause; to ski is the direct object of the past perfect progressive had been trying]
Here are two examples of an infinitive as a complement:
To bodly go where no man has gone before, one has got to be prepared [subject is one; object is to be prepared; complement is to boldly go where no man has gone before]
To eat spicy food, you've got to have a tolerance for burning sensations. [subject is you; object is to have a tolerance; complement is to eat spicy food]
If it can be replaced with another noun or adjective, including gerunds, an infinitive can be used.
Oh, and:
At least I know how to use "can't help" from this song " but I can't help falling in love with you...."
You could rewrite that as I can't help but to fall in love with you. But to fall in love and falling are both the objects of [can't] help. Sometimes you need to alter the syntax, though, as I've done in the above example. As long as you're using the correct syntax--and they're not used as verbs--you can usually substitute an infinitive for a gerund.
