My Spider idles well and it runs well above 1500 RPM. Idle when warm is 1000 RPM. When idling, if I slowly depress the accelerator, I am likely to stall the engine. If I hit the pedal hard, the engine responds quickly. Sometimes that will produce a slight hesitation before the engine races. When warm the engine is very drivable.

When cold, the problem is particularly bad. I have to keep the revs up to avoid stalling. When cold the idle is about 500 RPM.

One issue to consider is this car sits a lot, sometimes a month at a time. I use fuel preservative and sometimes add some carburetor cleaner (Seafoam) to the gas.

Do these symptoms ring any bells?

Keystone,
I'm assuming that your Spider is a carburettor model as '78 is probably a bit old for an injection model. I'm also assuming that it has automatic choke.

Cold idle....
When cold there is a bimetalic strip in the auto choke mechanism which closes the choke flap and engages the fast idle screw in the choke mechanism. The fast idle screw props the primary choke throttle flap open and controls the cold idle speed. When cold remove the air filter lid and check that the flap is operating. It should kick in when you first pump the throttle pedal. If the flap doesn't close then the bimetalic strip is either broken or badly adjusted. If it does close, check that the throttle is being held open. People often mistake the cold idle screw for the idle screw and tamper with it indiscriminately.

Stalling...
when idling the carb will draw fuel through the idle jet predominantly. If this was blocked you'd have trouble getting the engine to tick over. As the revs rise fuel will be drawn through the primary main jet. If this were partially blocked the mixture would be weak and the engine prone to stalling until the secondary choke opens. Dirt and debrise in the carb is quite likely with it standing.
However my money is on the accelerator pump diaphragm. There is a lever acting on the throttle spindle which pumps a squirt of fuel into the primary choke every time the throttle opens. The fuel mixture needs richening up in order to accelerate the engine revs and draw a greater charge when accelerating. Without that squirt the engine stutters and stalls. Once the revs have increased the greater air flow draws more fuel and the mixture stabilises so it only needs a brief squirt.
There are two common issues with the accelerator pump. The first being the diaphragm which creates the pressure can dry and split with age and lack of use. You might be getting very little "squirt" so tickling the throttle kills the engine but stamping it gets just enough to overcome the weak spot. Or the jet that points into the choke is held in by a hollow screw. The cap of the screw has a lead plug in it, these often pop out causing the petrol "shot" to trickle down the side instead of squirt into the choke.
With the lid off the air filter and the engine OFF (you don't want it to cough back in your face!) peer down the primary choke and operate the throttle gently. You should see the squirt and it should start immediately the throttle moves. If it doesn't squirt until part way through the flap opening then it isn't working properly.

Thanks, Spannerdude. You are correct about the carburetor. The choke is operating normally. It sounds like I need to have a carburetor rebuild kit installed which includes the diaphragm.