DSP on the Raspberry Pi
So, first of all, I'm biased. I maintain the GNU Radio source code.

So, what you want to do is pretty much perfectly fulfilled by GNU Radio


For flexible connection of signal processing steps, you'd use the "signal processing flowgraph composed of existing signal processing blocks" paradigm of GNU Radio, together with the excellent FIR filter blocks (maybe the gr_filter_design tool?) to ... filter stuff, the File Sink Block to save to file etc.
For real-time processing of high-bandwidth data (that means > 10 MS/s on a RPi3) you'd need CPU-optimized code: The current versions of GNU Radio use VOLK's optimized kernels; and thanks to Albin Stigö, these include optimized versions for the Pi's processor.


So, in my humble opinion, since you already seem to know what a FIR filter is: dive right into the GNU Radio guided tutorials; start with the "0th" chapter on why you'd want GNU Radio and work your way up. 

If you run Ubuntu or Debian in a current release on the Pi (not so much: Raspbian), you can easily install a very recent version of GNU Radio on that simply through an installation by apt.

Go wild! It's often a good idea to record and prototype your signal processing on a PC before moving everything to the Pi. GNU Radio makes that easy.