In English, if a verb is followed by another verb, those two verbs can be connected by 'to + infinitive', only the infinitive (omitting the 'to'), or the gerund (ing-form).
There are 4 categories of verbs:
(1) The ones that can use either 'to + infinitive' or the gerund, and it makes little to no difference, e.g. 'begin'. That means "He began to sing." and "He began singing." are both correct while showing little to no difference in meaning.
(2) The ones that can use either 'to + infinitive' or the gerund, but I makes a big difference/ changes the meaning, e.g. 'stop'. That means "He stopped to smoke." (= He interrupted an activity in order to smoke.) and "He stopped smoking." (= He quit; he doesn't smoke anymore.) are both correct, but do not have the same meaning.
(3) The ones that can only use 'to + infinitive', e.g. 'offer'. That means:"He offered to help him." is correct, while "He offered helping him" is not correct.
(4) The ones that can only use the gerund, e.g. 'consider'. That means "He considered going home early." is correct, while "He considered to go home early." is not correct.
Examples of verbs like 'begin' in (1):
begin, cannot bear, cannot stand, continue, hate, like, love, prefer, propose, start, etc. (there are many more; see lists available on ESL websites)
Examples of verbs like 'stop' in (2):
forget, regret, remember, try, go on
forget: I forgot to ask him. (= I didn't ask him because I forgot.)
I forgot asking him. (= I asked him, but I cannot remember.)
regret: I regret to tell you the bad news. (= I am sorry that I have to tell you the bad news now.)
I regret telling you the bad news. (= I am sorry that I told you the bad news; I shouldn't have done that.)
remember: I have to remember to call him (= something I have to do in the future, and I don't want to forget that I have to do it)
I remember calling him. (= I remember that I called him)
try: I tried to ride a bicycle. (= I tried, but I failed.)
I tried riding a bicycle, a tricycle and a tandem. I liked the bicycle more than the others. (= I tried out different options)
go on: After school, she went on to be a famous actor.
(= this is what she decided to do after finishing school)
He is a famous runner. He went on running until he was 86.
(= this is what he continued doing until he was 86)
Examples of verbs like 'offer' in (3):
agree, appear, arrange, ask, attempt, beg, cannot afford, cannot wait, care, claim, decide, demand, deserve determine, elect, expect, fail, get, guarantee, hesitate, hope, hurry, learn, manage, mean, need, neglect, pay, plan, prepare, pretend, promise, prove, refuse, remain, request, seem, swear, tend, threaten, wait, want, wish, would like, etc. (there are many more; see lists available on ESL websites)
Examples of verbs like 'consider' in (4):
admit, advise, appreciate, avoid, confess, defend, delay, discontinue, dislike, dispute, dread, enjoy, escape, explain, fancy, fear, feel like, forgive, give up, keep on, mention, miss, omit, permit, postpone, practice, prevent, recall, recommend, resume, risk, suggest, support, tolerate, understand, etc. (there are many more; see lists available on ESL websites)
!!! Exception !!!
Certain verbs are followed by the infinitive without 'to' when there is a direct object between the verb and the infinitive, and these verbs are: feel, hear, help, let, make, see, watch
Examples:
I can feel the temperature rise. / I can hear the bell ring. / I can help you succeed./ I will let him go./ It makes me laugh./ I can see him win./ We watched the tree grow.