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  and Digital Integrated Circuits  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 . 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 
(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  by Morris Mano
 https://www.amazon.com/Introduction-Microelectronic-Fabricat... https://www.amazon.com/Digital-Integrated-Circuits-2nd-Rabae... https://www.amazon.com/Digital-Design-3rd-Morris-Mano/dp/013...
 https://www.amazon.com/Digital-Integrated-Circuits-2nd-Rabae... https://www.amazon.com/Digital-Design-3rd-Morris-Mano/dp/013...