Is multicore programming the near-mid future of programming?

and if so, is Perl doing any effort (maybe trough Perl6) in that direction?

Yes. Perl 6 is specified with concurrency in mind, and there are some constructs that can be automatically parallelized, and many features are geared to make that painless. Sadly only pugs implements some parallelism so far, and only very basic constructs.

Will the functional programming languages become more relevant with the multi-core revolution?

Especially pure functional languages like Haskell. I can imagine it's much easier for the compiler to determine interdependencies than for normal, imperative languages. But it remains to be seen how much that will matter in practice; I believe that both language families have much space to improve wrt concurrency.

Will it worth to start today my project in one of those concurrent programming languages?

That really depends on how easy or hard they are to parallelize. I'm currently doing some simulations that are embarrassingly simple to parallelize - I always have to cover a large parameter space, so I just start one process per parameter, let it run for an hour and collect the result in a file - there's really no point parallelizing that more unless somebody throws more than 200 CPUs at me, each equipped with at least 8GB RAM.

Which ones will survive, which will be abandoned and which will be a reference for concurrent programming?

I don't think that's clear yet. There are a few language that I'm pretty sure will stay, which have proven to fit very well for some problems (C and Haskell for example; I'm also pretty sure that Erlang has a sufficiently large community to survive).