Gwen Whiting, co-founder of The Laundress has a degree in textile science from the Ivy League, and was actually graded for ironing. Her advice is below.
Step #1: PREP
Check for grime on the bottom of the iron. Also clear the ironing board of any debris. If it's really dusty, throw the cover in the wash.
Step #2: START WET
Skip the dryer, ironing straight from the washing machine. Not possible? Use a spray bottle of water to dampen.
Heat: Crank iron to whatever temp matches your shirt. Look for the itty-bitty words on the dial and on the back of your shirt tag. If your shirt is oxford cloth, crank to "cotton/linen."
Step #3: STARCH?
It's optional and builds up over time, so you should occasionally throw the shirt in the wash to remove build-up, even if you usually dry clean.
Types: Corn starch is for natural fabrics; sizing for synthetic fabrics. Spray on before ironing begins.
Step #4: COLLAR & CUFFS
Collar: Pop it and iron from the tips toward the middle. Iron the inside. Flip. Do the outside. Don't turn down the collar until the rest of the shirt is ironed. Do NOT iron a crease into the collar.
Cuffs: Starting on the inside, iron from bottom edge toward the sleeve. Flip cuff. Repeat. Also poke the tip of the iron into the pleat(s) just above the cuff.
Step #5: SLEEVES & YOKE
Sleeves: Hold up and tug taut the arm so you've got a crisp straight fold from shoulder to cuff. Lay sleeve on the board, and in long sweeping strokes, iron in a straight solid crease. Do the back of the sleeve first because inevitably you'll get creases, so save the front for last. Slide the armpit part of the sleeve over the tip of the ironing board, and iron flat the shoulder.
Yoke: Staying in that position, hit the yoke, that double-layer strip that connects the collar to the shirt body. Swing the iron from shoulder to mid-back. Switch shoulders. Repeat.
Step #6: TRUNK, BACK, FRONT & DONE
Front non-button side: In long strokes from collar down, start with the placket (the strip with all the button holes).
Pocket: Iron from the bottom up.
Back: Iron below the yoke, from top to bottom.
Front button-side: Lastly, using the tip of the iron, weave in and around the buttons.
Done: Hang it up so as not to have to do it again (wooden hanger preferred).
Do you iron at work?