There are several ways of going about it. Here's what I did and why-
1. good scope (tek 465 with 100 MHz BW and x-y capability)-- to visually center and make symmetrical the FM detector transfer function, plus use it to synch one waveform with another (19 khz pilot for instance). Also used to observe signal flow in case there is a problem with the unit besides being out of alignment.
2. good dvm
3. good fm signal generator to provide unmodulated and modulated RF to the receiver front end. I use a SG165 Sencore, but the ST1000A is better. This stereo generator needs to provide unmodulated, L/R mono, and LR stereo MPX signal to RF modulator. Also, tunable over the broadcast band is best, and modulation (composite to RF) linearity is important for analyzing tuner distortion and adjusting it to min value. Also does not hurt if you have a way to verify % modulation, if it is not calibrated, and a way to measure or understand the RF level being output, to see if you are in the ballpark (+-1.5 dB or so) with sensitivity specs.
If this generator has a 10MHz I/F sweep mode, then you can optimize the alignment of the I/F stages.
4. A low noise (distortion) sine wave oscillator to drive the stereo generator. I modded the SG165 to take and external source (Wavetek 180). The wavetek worked well after being on for about a month driving a 50 ohm load, then going through the sine wave distortion cal on it using the HP LF spectrum analyzer to minize the 2nd and 3rd order products. I run it at 1KHz. If I change the output level a lot or the frequency from 1Khz to 10KHz, I have to re-optimize it, since it is more a function generator than a sine wave generator. But I can keep harmonics down to below -70dB with it.
5. A way to measure the audio distortion out of the FM detector. I use an HP 8556A LF plug in for the HP141T spectrum analyzer for this.
6. A way to look at the raw RF coming into the tuner under test, before and after amplification (to measure levels and modulation index), the noise of the local oscillator (for grins mostly), and the I/F chain envelope. I use an HP 8553B spectrum analyzer RF plugin for this.
7. Good resolution frequency counter. I found an HP 10 digit counter, oven stabilzed, at a local ham meet. It is not sensitive enough to measure RF freq directly from the SG165, but it can measure the output frequency of the I/F, the LO, and the RF after single stage amplification. I use it with the tracking generator function of the analyzer, to know what freq value I'm looking at on the display, since it is old school, and does not annote this on the screen. I also use it to set 19 KHz MPX functions when required.
8. A bunch of bnc cables, adpaters, and loads to terminate things properly.
That's a lot of stuff that has to work properly to fix or optimize a tuner needing assistance, but it is an addictive hobby, more so than recapping units...
There are some good textbooks on the subject also, but I don't have any links handy.
Get a cheap GW tuner from the '70s to start. Get a couple more, and then you will be sucked in.