When I first learned English, I was taught to use only "it ~ to" form as opposed to "it ~ ing" form, so I always thought the latter one is not grammatical. But I heard a lot of native speakers actually using the second form, so I wondered if it's grammatical or it's ungrammatical but allowed in daily conversations.
For example:
It was the first time to meet you.
It was the first time meeting you.What is it like to have a child?
What is it like having a child?
From M.Swan's PEU:
Infinitives are forms like (to) write, (to) stand. Unlike verb tenses (e.g. writes, stood), infinitives do not usually show the actual times of actions or events. They usually refer to actions and events in a more general way, rather like -ing forms. Besides simple infinitives like (to) write, there are also progressive infinitives (e.g. (to) be writing), perfect infinitives (e.g. (to) have written) and passive infinitives (e.g. (to) be written).
An infinitive clause can be used after be as a subject complement.
Your task is to get across the river without being seen.
My ambition was to retire at thirty
Sentences like these can also be constructed with preparatory it.
It is your task to get across the river without being seen.
It was my ambition to retire at thirty
We can use -ing forms (e.g. smoking, walking) not only as verbs, but also like adjectives or nouns. When -ing forms are used as verbs or adjectives, they are often called 'present participles'. (This is not a very suitable name, because these forms can refer to the past, present or future.) When they are used more like nouns, they are often called 'gerunds'.
We can use it as a preparatory subject or object for an -ing form. This is usually informal.
It's nice being with you.
I thought it pointless starting before eight o'clock.
This is common with any/no good, any/no use and (not) worth.
It's no good talking to him - he never listens.
Is it any use expecting them to be on time?
It's no use his/him apologising - I shall never forgive him.
I didn't think it worth complaining about the meal.
The second form is grammatically correct. The word ending with "-ing" is a gerund (while the word corresponding to "to" is an infinitive). Both infinitives and gerunds are nouns, so they are equivalent grammatically.
Which form is used in speech is down to preference, and, as you noted, the second form is more common.
Let's get it straight, your first pair of sentences is a complete mess!
Neither the option with the to-infinitive nor the option with the gerund really work here since the sentences are illogical and ungrammatical.
"the first time to+verb" is plain wrong (bad English) whether in the past or the present, or the future.
However, there seems to exist a construction "the first time to+subject" as in "It was the first time to Jupiter. We didn't know what awaited us there.". Guessing the meaning here is "visit/journey/trip/travel". However, as most sources say we need a possessive adjective with "first time". So your correct two sentences should be:
Notice, that the to-infinitive is stiffly formal. And by what I know the first sentence means "something that was about to happen" while the second one means "something that happened". However, in casual speech it's usually:
With the second pair the difference is vague but grammatically interpreted.
According to Cambridge Dictionary - "Verb patterns: verb + infinitive or verb + -ing" some verbs can be followed by either a to-infinitive or -ing the "difference in meaning is often small" (quote from the reference):
However, here we can see a question form ("What is... like") which we use when we are enquiring about experience. With this question form either the to-infinitive or the -ing will both mean the same thing. Although we could interpret them as with the verb "like":
Based on this information we get:
To my own knowledge the -ing form usually conveys the idea of wanting to get into somebodies skin. To experience what they experience!
