It's very simple. Put motor oil in the engine, and universal hydraulic fluid (e.g. New Holland 134) everywhere else. There are three reservoirs besides the engine: transmission, hydraulic and differential. All three have large filler plugs you can easily find on the top of each unit. The transmission and differential have level plugs on the side, about one-third of the way up from the bottom, and the hydraulic reservoir has a dipstick. The dipstick probably has different markings on each side; use the side that says "hydraulics", not "pedestal". (The pedestal side is used to check the oil level in the row crop steering pedestal.)
The original recommendation was to use 80W gear oil in the transmission and differential, but using universal hydraulic fluid is simpler, and (in my opinion) better.
There is also a steering gearbox at the bottom of the steering column. Although it originally held gear oil, it's probably been filled with corn head grease so it won't leak.
Things get more complicated if you have a row crop tractor, but it wouldn't be a 640 if you did. Row crop has two final drives plus a steering pedestal, all of which use 80W-90 gear oil.