When contemporary fundamental logical notation was established? I mean basic symbols as used nowadays $\iff\implies\land\lor\lnot\forall\exists\vdash\models$.
The answer is different for different symbols. It wasn't invented all at once, but accreted gradually. Even closely-related symbols like $\land$ and $\lor$ or $\forall $ and $\exists $ were introduced separately. (Strange but true.)
$\lor $ dates to the 19th century and is an abbreviation for Latin vel. $\land$ didn't come until the following century by analogy with $\lor $ and with $\cup $ and $\cap $.
$\exists $ was invented by G. Peano for his Principles of Arithmetic (1889), which was enormously influential, and also probably the last major scientific work to be written in Latin. It also introduced $\supset $ for implication, originally an upside-down capital ‘C’. Gentzen introduced $\forall $ in 1935 by analogy with $\exists$ in Untersuchungen über das logische Schließen ("Investigations on Logical Reasoning").
$\vdash $ and $\lnot $ are abbreviations of the elaborate notation of G. Frege's Begriffsschrift. $\vdash $ was introduced in Principia Mathematica. $\vDash $ came later, analogous to $\vdash $. $\sim $ for logical negation is much older than $\lnot $ and is an altered letter 'N', but Hilbert used it to mean logical equivalence instead of negation. Gentzen attributes $\lnot$ to Heyting.
$\to $ for implication is ancient, going back at least to Boole. I don't know when $\implies$ was introduced but it will be hard to trace since even today the distinction between $\to $ and $\implies $ is not standard.
If I were going to give a short summary, it would be that it came together in the early part of the 20th century, based on equal parts Peano, Frege, and older traditions, modified by Principia Mathematica, and had settled down by 1930 or so.
The Van Heijenoort sourcebook would be a good place to start looking if you wanted to follow up in more detail.