Firstly, you need to decide on what it is that you want to measure and why.
I measure amplifiers both for a bit of fun (yes I know, I should get out more, but that's currently not possible.....) and when I've repaired something to make sure it's working correctly.
Let's assume for the sake of this post that your interest is in working out if the amplifier is working correctly, not because you're designing amplifiers or verifying production.
In order to test an amplifier you'll need the following:-
A source of signal with very low distortion, from, say 1mV up to 10v and from 10Hz to 100kHz. Square wave option is also useful.
A millivolt meter that can measure audio levels from -100dBu to +30dBu ( 15uV to 50v) unless you're going to be testing some ultra-high power amps. It should have a bandwidth of 100kHz and it would be useful if that could be switched to 20kHz and have some standard weighting filters.
A distortion meter that can measure down to 0.01% is sufficient to make sure the amplifier will be completely transparent. It might not be enough to verify if the amplifier fully meets its specification, but my view is if it's so low I can't measure it, I don't need to worry about it.
An oscilloscope so you can see what's happening. A 20MHz bandwidth is sufficient. Dual trace is very useful indeed, almost obligatory.
A dummy load that can be set to 2, 4 & 8 ohms, of sufficient power handling for the largest amplifier you'll ever want to test. Having two will allow you to do 'both channels driven' testing. It's also useful to have a loudspeaker simulator load that provides some reactive loading, but there's no standard for that, so comparisons with other tests will be pretty meaningless. I use 8 ohms in parallel with 2uF as my standard, especially for square-wave tests.
Some of those instruments can be combined, and a lot can be done using a high quality PC sound card and some free software like REW. The one limitation of PC software I've found is that none I've come across measures distortion by nulling out the fundamental and displaying the residual. All do it by FFT, which is a much more powerful technique, but doesn't show things like crossover distortion as obviously as looking at a distortion residual on a 'scope.
You'll also need a set of interface cables for ins and outs, a load of adapters for the different sockets used and plenty of patience.
Good luck. Measurements can be a fun hobby in its own right.
S.