IMO a good way to start is by buying an FPGA development kit such as https://canada.newark.com/terasic-technologies/p0037/dev-boa... which has a bunch of I/O and cool things to do with.
The FPGA kit will come with some golden designs and instructions to download and install the EDA software. Feel comfortable with the FPGA software and make some leds blink..
Now you're in a position that you can download a hardware design to the FPGA and use the pins!
RTL Hardware Design Using VHDL: Coding for Efficiency, Portability, and Scalability is an amazing book for learning how to do RTL logic, buy the book, read, execute the examples!
Now you have a kit and you know RTL! Next you can learn the algorithms you want and implement them!
My workflow used to be something like:
1. Implemented an algorithm using fixed point math in Matlab (or Octave)
2. Develop the VHDL equivalent to (1), test with something like ModelSim
3. Generate the RTL with the EDA software and download to the board!
To test step (3) it really depends what your algorithm is.. if you can easily control your inputs and outputs through a computer connected to the FPGA then it is a good way to go about it! You might also want to develop a RTL module to connect to your design and generate the inputs / see the output, or buy some expensive hardware to verify that your design is working as intended.
Hopefully that is helpful.. it is a vast field and there are many algorithms, FPGA sizes and etc. A kit like the one I sent should be enough to go from 0 to a decent size design...
I was about to write a thorough answer but jsolson's very good answer mostly covered what I had to say. I'll add my 2 cents anyway
- To understand the manufacturing process you need to know the physics and manufacturing technology. That's very low level stuff that takes years to master. There are books and scientific papers on the subject but it's a game very few players can play (ie. big manufacturers like intel and TSMC, maybe top universities/research centres). You can try some introductory books like Introduction to Microelectronic Fabrication [1] and Digital Integrated Circuits [2] by Rabaey if you're curious
- You can design ASICs without being an "expert" on the low level stuff, but the tools are very expensive (unless you're a student and your college has access to those tools). You need to learn VHDL/Verilog, definitely need the right tools (which, I'll say it again, are too expensive for hobbyists) and extra money to spend for manufacturing.
- FPGAs are different. No need to know the physics and you don't have to bother with expensive tools and foundries, the chip is already manufactured. My advice is to
(a) get an FPGA board. Digilent boards are a good choice as jsolson said, but there are other/cheaper solutions as well [3][4]. You'll still need software tools but (for these boards) they are free
(b) learn VHDL and/or Verilog. Plenty of books and online resources. A good book I recommend is RTL Hardware Design Using VHDL [5]
(c) I assume you know some basic principles of digital design (gates, logic maps, etc.). If not you'll need to learn that first. A book is the best option here, I highly recommend Digital Design [6] by Morris Mano
[1] https://www.amazon.com/Introduction-Microelectronic-Fabricat...
[2] https://www.amazon.com/Digital-Integrated-Circuits-2nd-Rabae...
[3] https://www.scarabhardware.com/minispartan6/
[4] http://saanlima.com/store/index.php?route=product/product&pa...
[5] http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0471720925...
[6] https://www.amazon.com/Digital-Design-3rd-Morris-Mano/dp/013...
High Speed Digital Design: A Handbook of Black Magic by Johnson and Graham:
http://www.amazon.com/High-Speed-Digital-Design-Handbook/dp/...
and RTL Hardware Design using VHDL by Chu:
http://www.amazon.com/RTL-Hardware-Design-Using-VHDL/dp/0471...
(the latter for designing for FPGAs)