You can probably work on design for FPGA's for free. I have a student license of Vivado for VHDL/Verilog. It allows for synthesis, analysis, and deployment to certain boards.

ASICs are another step trickier. I would think that you can't actually manufacture a chip without paying some huge cost. For commercial applications, I think you would need to pay for some licensing.

Given the costs involved, practicing core design on FPGA's might allow a stepping stone to where you want to be.