Voltage and amperage relations

If i have 12V/3A power supply and step it down with LM2596 to 5V, what amperage i can expect? Will i can still have 3A?

It's rated for 3A, so with the correct choice of inductor and a good layout it should be possible.

With a proper step-down converter, you can get 6A at 5V. Designing that yourself with just the chip and the datasheet is much more complex than the datasheet would suggest.

Buy a module off the shelf, unless you just want to learn while blowing up valuable components.

What are you trying to do?