With the exception of power and ground, there is no pin on the Arduino that can require current that exceeds what can be handled by those ubiquitous cheapo jumper cables. Those pins are connected directly to the pins on the Arduino, and are recommended to source or sink no more than 20mA, 40mA absolute maximum - which the jumper cables can handle no problem.
If you need to handle higher current you need an external switch (typically a MOSFET), and you'd only need to make the other two wires going to the switch thicker.
The connectors that work are called Du Pont 0.1" connectors. They're very common. You can crimp them yourself, with a crimp tool, but it's very unpleasant (though better than it was before someone posted a guide in the electronics section). IME, male connectors going into female header doesn't work well, doesn't make good contact - you get a much better connection with male pin header and female jumpers, and from male pin header and female pinheader.