How do pilots start their plane's engines? Is there like a key that they have to turn to ignite the spark plug in the engines or is it much more complex?

Depends on the plane.

I mean if we are talking an ultra light ran by a lawnmower engine they have to get out and pull the cord.

If we are talking about jumbo jets there are a lot of toggles and buttons that have to be in flipped/pressed usually with a final button to press that starts the engine. Usually all of those other buttons control secondary systems, control fuel, control de-icing, blah.

Private prop planes are more like a car in that there is a key, usually unlocking the control panel allowing the press of a button to run a "starter" to start the plane.

Keys are usually to unlock the controls/start mechanism in planes. Keys in cars usually are wired to the "starter" an electric device that turns the engine over to start it up plus they unlock the steering wheel and may also be used for alarm systems.

Older prop aircraft need to have a switch "turned on" and somebody outside to turn the prop to get it to fire. Some even had to be loaded with a gun-like cartridge that had gun powder in it to give the engine its first "kick" revolution to start.

I can't cover all of the examples.

Smaller planes have keys. Bigger planes have switches and levers.

In the attached screen shot from FlightGear, you see a mockup Cessna 172 cockpit. The starter is in the lower left, with settings Off/1/2/Both/Start
1, 2 refer to the two redundant magnetos (ignition). You nudge to start until the engine runs.

The pilot gets out, grabs the propeller and gives it a strong heave, then gets in as quick as he can. ;)

It's a little bit more complex than simply turning the key, but for typical single-engine light airplanes, it does involve a key.

The first thing you generally do when you get to the engine start checklist is make sure the fuel gauges read properly. If necessary, you'll prime the engine, putting a bit of fuel into the cylinders so that it starts more easily. You'll make sure the mixture control is set for full rich to deliver maximum fuel to the engine during startup. You'll then crack the throttle open a bit. Then, you'll make sure the area around the propeller is clear. Finally, you'll turn the key to the "start" position, releasing it once the engine catches.