First, the older 8 bit boards are woefully under-powered for this sort of project. Plan on using 32 bit hardware, at least ARM Cortex M3 or M4.
Personally, I don't believe in the philosophy expressed on this thread, that this stuff is so hard and can't be made accessible for beginners. Why? Well, because I've put quite a lot of work into a large audio library over the last 4 years. It has 2 types of tone detection, plus 1024-point FFT analysis, which really are easy to use. The library runs on Teensy 3.2, 3.5 & 3.6.
Here's the library on github:
A few years ago we made a tutorial for getting started. The tutorial was written before either of the tone detection features, but it does cover FFT and the general way you use the library. There's a 31 page PDF and a 45 minute full walkthrough video, in case you get stuck and want to see how to do any part.
Usually FFT frequency bins end up being too coarse if you're looking for very specific frequencies. If you use the audio library design tool, scroll to near the end of the long list of audio features. In the "analyze" section, you'll find "tone" and "notefreq".
The "tone" analysis looks for the amount of a single (or narrow bandwidth of a) frequency. It's actually using the Goertzel algorithm. You can configure how long the analysis computes, where longer times give higher selectivity, but slower response. Like everything in the design tool, just click on the object you want and the right-side panel updates with the documentation for the functions you can use in Arduino to control it. There's also a couple examples (in the libraries File > Examples > Audio menu) for using 7 of these tone detection objects to decode DTMF dial tones.
The other one is "notefreq". It uses the very advanced YIN algorithm to search for the strongest fundamental frequency. Unlike "tone" analysis, where you choose the frequency, it searches for the frequency and tells you what it finds. Usually the main application is recognizing musical notes. It works very well, even on complex sounds like guitars and tubas. It also consumes a lot of CPU time, nearly all the power of a Teensy 3.2, so if you're going to use this and want to also have FFT or other computationally heavy stuff, you'd probably want to use the faster Teensy 3.6 board.
As you can see in the example and design tool documentation panel, these fairly advanced DSP features come packaged in a library that's very easy to use and you can access their results from Arduino sketches using the familiar available() and read() functions, to know when the analysis has produced more data, and to read it into your code.
Sure, inside the library is rather complicated DSP code. But just like you don't need to be an expert mechanic to simply drive a car, you really don't need an advanced engineering degree specializing in DSP to simply use these powerful analysis tools. All you really need is a good library. That's what I've tried to make for you, and for everyone who might need to do this sort of sound processing. Hope it helps?