Forget ASIC design, unless you want to get a 2nd mortgage to license the tool chain for one year. Plus, mask sets will run anywhere from $10,000 to $1,000,000 (depending on the geometry) to get the chip produced.
Learning an FPGA is actually surprisingly attainable. You can get many boards with smaller parts for under $100. The tool chain is free, but you are stuck with proprietary software.
My own experience is with Xilinx, but they recently went to a new "Vivaldi" software suite that supports the newer chips. Older chips are stuck using "ISE" software, which does not run on Windows 8 and up without hacking (yuck). So, if you buy Xilinx, make sure that you get something supported by Vivaldi. I understand that Vivaldi also supports SystemVerilog -- VERY nice to have for testbenches, but not a lot for RTL code. Altera is also VERY popular and worth a look, and I believe that Lattice and Actel might still be in business.
Next, you need to learn RTL (register transfer language) -- VHDL or Verilog. Both have their pros and cons, but I prefer Verilog. It is very much like programming in something like "C", but every "always" block runs CONCURRENTLY! In other words, all code runs at the same time. This makes sense because all transistors are running at the same time. There is a web site called "World of ASIC" that has some nice tutorials.
I would also check the "hack a day" web site. They had links to tutorials using a $20 board a few weeks ago.