There's exactly one formula you need, and one mnemonic that makes it stick for life: F = C × 9/5 + 32. Everything else about temperature conversion is a variation on it.
The three equations
Celsius to Fahrenheit: F = C × 9/5 + 32. Fahrenheit to Celsius: C = (F − 32) × 5/9. Celsius to Kelvin: K = C + 273.15. That's every temperature conversion you'll ever need. The 9/5 comes from the fact that 180 Fahrenheit degrees span what 100 Celsius degrees do (between freezing and boiling), so each Celsius degree is 1.8× a Fahrenheit degree. The 32 is the offset between zero points.
Why it's "affine", not linear
Most unit conversions are linear: 1 meter = 100 cm, 1 kg = 2.205 lb. Multiply by a constant and you're done. Temperature is affine — linear plus a constant. Because Celsius zero and Fahrenheit zero are different physical points, you need both a scale (9/5) and an offset (+32). This is why "twice as hot" doesn't work the same in the two systems.
The useful shortcuts
For quick mental math when reading weather, (C × 2) + 30 ≈ F. 20°C → ~70°F (actual: 68°). 30°C → ~90°F (actual: 86°). Good enough to know if you need a coat. Don't use it for oven temperatures — the error compounds at the top of the scale.
The only exact crossover: −40°C = −40°F. If you see that number, you don't need to convert.
C, F, K — one click, shareable URL. For when the shortcut won’t cut it.