How the color wheel actually works
There is more than one colour wheel, and most arguments about colour come from two people using different ones.
#ff0000
#ffa500
#ffff00
#008000
#0000ff
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Two wheels, two jobs
The wheel taught in art classes has red, yellow and blue at its corners. It describes pigment: what happens when you mix paint, where each addition absorbs more light. The wheel inside design software has red, green and blue, because a screen adds light instead of subtracting it. Neither is wrong. A painter mixing blue and yellow gets green; a screen averaging blue and yellow light gets grey. Ask which material you are working in before you ask which colours are opposite.
Hue, saturation, lightness
Position on the wheel is only the hue — the name of the colour. Two other numbers decide how it behaves. Saturation is how much of the colour there is, from grey at zero to the purest version at a hundred. Lightness runs from black to white through the hue. Most palettes that feel wrong have the right hues at the wrong lightness: five colours all at the same lightness leave nothing to read against, however pleasing each one is alone.
Warm and cool are relative
The wheel splits roughly into warm colours — reds, oranges, yellows — and cool ones — greens, blues, violets. But temperature is comparative rather than absolute: a blue-leaning red is cool next to a orange-leaning red, and a green can be the warm colour in a palette of blues. What matters in practice is consistency. Mixing temperatures without meaning to is what makes a set of otherwise good colours look accidental.
The part to remember
Decide whether you are mixing light or pigment, then treat hue, saturation and lightness as three separate decisions rather than one.
Mix two colours and watch the wheel work