I was in a very similar situation about a year ago, and I now have a flying aircraft. Below is what worked for me, but many other routes are possible.
First I purchased an almost ready to fly quadcopter kit from a well-known manufacturer. I only had do a bit of soldering, assemble the frame, configure the flight controller, and bind the transmitter. While this step may not seem necessary, it allowed me to develop a basic understanding of how the system works as a whole. I had no prior RC knowledge. It also allowed me to learn how to fly, and to have realistic expectations about how a well-tuned quadcopter should behave.
Then I got an Arduino and basic sensors. At this point I was not sure I would end up building my very own quadcopter. I just wanted some toys to play with and see what I could do without spending too much money. I am a hobbyist programmer so the C++ part was not a problem. You are a web developer so you will have to learn how to program real software ;-) This step allowed me to confirm I could connect a microcontroller and a few sensors without blowing up the house. Eventually I ended up with a working inertial measurement unit (IMU), combining gyroscope accelerometer and magnetometer data into an accurate representation of the device orientation in space.
It then became clear that building a quadcopter was a realistic target. I spent a little more money to purchase motors, propellers, ESCs, a radio receiver (I already had the transmitter from my first quadcopter) and a few wooden sticks for the frame. With some more programming efforts I eventually got everything to fit in together nicely.
The whole process took me about a year and I'm still working on it. It's worth noting that along the way I also learnt about a few unnecessary things, like servo motors, barometers, and sonars. I did not mean to take the fastest route. If like me you have a day job and a life this is at least a 6 months long project. In order to keep me motivated I posted my steps on a blog and I only set myself realistic targets: I did not start thinking I would build a quadcopter, instead I thought I would learn the Arduino basics and refined my goals as I progressed.
I went to uni (10 years ago) and I had a basic level in physics, mathematics, and electronics. The difficult skills I had to learn were 3D vector geometry and controller theory. 3D geometry is for the IMU. I used quaternions because Arduino requires optimisations for speed. Matrices and a better microcontroller is probably a more reasonable choice. Controller theory is to implement and tune the PID controllers which drive the motors so as to keep the aircraft level. These are all skills that you can learn online. I never bought a book for this project. I also didn't model my quadcopter mathematically like you seem to suggest.
This is my experience, I hope it is helpful to you. I'm sure the challenges will be different for you anyway. If I had to summarise I would urge you to purchase some components/parts and see what you can do with them because learning quadcopters is more about practice than theory. Good luck!
