Priorities
I think Mechanical Mandy pretty much covered the range, so I will not repeat.
Instead, when I was looking at the beautiful 58/59 Mulitimatic Thumper my partner got last month, my priorities were divided among these areas:
Safety:
As anyone who ever had a rollermatic go up in flames knows, Frigidaire used some synthetic rubber parts which DOW couldn't have improved on. So I made sure there were no mechanical friction or binding problems anywhere in the machine. No pinched hoses, etc.
Checked the electrical situation before plugging her in. Any shorts, open circuits, grounds (o ye gods and little fishes, no ground...took care of that...got all the metal components bonded and the machine properly hooked up to a heavy copper three prong Frigidaire cord from a later machine). The cold water hose was not the original, no grounding wire or braid in it!
Replaced the simple water valves with new ones. They aren't thermostatically controlled three-ways, just plain. In any machine this old, they will cause problems and flood you sooner rather than later. You must pay attention to the grounding here, new ones are plastic clean through and do not permit grounding through the hoses. Nor do the new hoses which are also a good idea.
Mechanical:
Carefully tested the mechanical systems. No grinding, crunching, bad sounds - no loose bolts or parts, out of this world rust or otherwise. Did not lube anything, all the rollermatics of my misspent youth are still whirling in my mind. Did check the belt tension, was relieved.
Added hot water and let the seals come up to temp. Checked for leaks everywhere - as Robert once mentioned, a very small drip through the bellows and the bearings seize faster than you'd have thought possible. Gently tried agitation and pumping. Since that was ok, let 'er rip.
Oh, ok, two small things occur to me which you need to look at. One, I speak lovingly about "thumpers" but in reality, they are quiet. If you are getting some thumping, as in plastic on plastic, then you need to make sure the pulsator spacers have not perished. It doesn't take much to kill one and they aren't exactly being made by the million any more.
Two, as durable as that timer mechanism is, (and was until GM switched to the built in obsolescence of the driver dogs in the timer motor) the motor reset switch has a very tight clearance to the panel. Be sure it can move enough to reset, otherwise you may be left sitting there with an otherwise fine machine you can not get to work. Shorting that control is so not a good idea...