I am reading now a book about Russian language (The big silver book of Russian language p. 2) and it's written there that when people want to talk to new people then they should say the sentence: "Давайте будем на ты". I've never came across it. Is it indeed done?
ты is the informal usage for you. It is used with family
members, children, close friends, and pets. When Russians wish
to speak informally to one another for the first time, they
say, “ Давайте будем на ты .”
The Russian has two forms for you. They vary according to
formality and number.Тыis informal singular, вы is formal singular
and both formal and informal plural. Вы is the marker of formality,
politeness, and seniority. Whenever you are in doubtabout which form
to use, err on the side of caution and use the polite form
(Вы) unless you are speaking to a child or a pet.
Yes. "Давайте будем на ты", is a well known and standard phrase. You can also say "давай будем на ты" to emphasize the fact that you are very inclined to be less formal.
The other standard forms are: "Давай[те]/Может [сразу]/[перейдём] на ты?" or "можно на ты [сразу]/[уже]?"
It's worth to know also the verb "тыкать". It's a colloquial term for referring someone by "ты". You can here something like "Ничего, что я тыкаю? Мы ж всего день знакомы". The verb "тыкать" has homonym which translated as "to poke" and should not be confused with.
To add to the already posted answers, I'd like to point out that the statement that
when people want to talk to new people then they should say the sentence ...
is not exactly true. It sounds as if your book suggests that whenever people talk to new people, they should say this sentence — but that isn't so. You can never ever say this when you've just met new people and started talking to them, because that would be rude. It's only appropriate when you've been communicating at least for some time, and you feel that you and your vis-à-vis have established a certain rapport that allows for less formal relationship.