Just went through this on a 300 Utility that had sat for 6 years.
Here's my checklist for starting old cars and equipment:
check oil/fluids. look for obvious faults like broken belts, leaks, bare wires etc that may cause trouble later on. Keep an eye out for levers/sharp stuff/rotating objects that you may not be familiar with on a new-to-you machine. A forklift I just bought this week had some bare wires hanging loose under the dash. They had never been a problem for the previous owner since it was always driven on level ground...but the first time I drove uphill with it, they swung back into the dash, made contact and nearly started a fire!
Put trans in neutral and set brake/block wheels
Check air cleaner for water/mouse nests/debris
install fresh battery and check that the engine cranks with the key on
check ignition - is there 12v present at the coil positive terminal when the ignition is on? Remove cap and inspect points for voltage or spark when opened and closed (note - make sure points aren't being held open by distributor lobe. If so rotate engine until points are closed, then manually open them). An alternative is to pull the coil wire out of the dist cap and place it close to a clean grounded surface - within 1/8" or so - and crank the engine. You should see spark. Points are almost always oxidized after long dormant periods and should be cleaned.
Fuel - is there decent fuel in the tank? Is the sediment bowl clean and does the valve work? Shut off fuel, remove the sight bowl and clean out, then open stop cock and let a little fuel run out to make sure the inlet isn't plugged. Replace bowl and re-open valve. If no fuel flows, you might have to remove the whole shutoff assembly and inspect for blockages. Sometimes crud blocks the opening and simply inserting a probe or small screwdriver into the fuel tank will open it up.
Check for fuel at carburetor - either pull the carb drain plug or disconnect line. There is often a small mesh fuel filter just inside the carb where the line attaches. it gets clogged easily. You'll want to see a steady pencil thick stream of fuel. An inconsistent dribble means a blockage somewhere.
In the case of the old 300 clean fuel dribbled out of the float bowl drain plug, but the bowl was full of hard caked rust. Water must have gotten in the carb at some point. It required removal of the bowl and cleaning all the passages with carb cleaner and compressed air. Even then it still took quite a bit of tinkering and several heat cycles before it was running steady and reliably.
Hope that helps.