I was reading your post and immediately thought of the Teensy’s as well, and SPI, so basically I strongly agree with everything dgsharp said. That is certainly a very doable way to go.
Look at the PJRC.com website, you will see a lot of data on the teensys, and look at the signal processing that Paul has already done with his Audio board and library for the Teensy 3.1, I think it also works with the 3.6, (I haven’t checked that,) but even if the hardware is not appropriate for you, the libraries are a very good starting point for FFT analysis etc.
Even if you are new to soldering, spend the money for a cheap temperature controlled soldering iron with replaceable tips. Not the big-tipped plug it right into the wall type that just tell you the wattage! I never go over 650F on a circuit board, and that only for desoldering, 600F is good for SMT soldering, no more than that. AllElectronics.com has their IR-50 and with the 2mm chisel tip, that would be a pretty good setup for surface mount soldering. For SMT, I use a Hakko T18-CF1 1mm Bevel Tip, but that’s a few levels up in price) Look online there are a lot of good videos on how to solder surface mount chips. Make sure you also get good quality solder wick (I prefer MG Chemical, but there are others, but don’t use All Electronics solder wick, it’s not terrible, but it’s worth paying for better) maybe one to two mm would be best for surface mount fixups. You should also have a magnifying glass and I’ve found no-clean liquid flux very helpful. An inexpensive small, nylon bristle paintbrush is helpful with the flux, or I use the Haiko brush flux applicator. The biggest mistake people make is using too much solder, but with the wick that can be fixed, I mess up with too much solder all the time, a light touch with the solder wick, and usually the pin is perfect… Look closely, at each pin and make sure they look bright (if not, there might not have been enough flux) and that the solder is not balled up, indicating a cold joint. Both issues easily fixed.
There is one trick I use that makes the wick even better. If you do get no-clean flux, brush a little on the wick wider than the area you are trying to clean up, it makes the wick that much better. But If cost is a concern or you don’t plan on doing more than this one project, then a good quality wick alone is probably fine.
Get decent quality breakout boards for the chips you are trying to use, SchmartBoard are good, but not the only ones, and plan on messing up one and you should be able to SMT solder with no worries. If the chips you want to work with are expensive, find some chip that is cheap with the same pin spacing, (doesn’t even have to be the same number of pins,) and practice on one or two of those. For more of a challenge, with the practice ones, see if you can get the chip back off, clean up the board and do another chip, (or the same one!) on the same board. (hint, use a heat gun or hair dryer to get the whole breakout board as hot as possible first, or even carefully heat the board in a flat bottom pan in the stove. Definitely try that only on throw away parts until you get the hang of it.
The main point is, don’t be afraid of soldering, don’t use too much solder, and practice a bit first…