There are actually four methods for ironing a men's dress shirt. They should be attempted in this order until all wrinkles are removed from the garment.
Method 1: Ask your girlfriend to do it for you.
The key here is to ask nicely. Avoid a tone of voice that says "of course you should iron my shirt simply because you are a woman". Try to communicate that you are just a man in dire need of a skilled female touch. Remind her how much you appreciate her help and and celebrate her skills. It's what makes you a great couple. She iron's the shirts, cooks the food and cleans the house. You kill the mice, fix the lightbulbs and... other stuff...
Yeah, I know, sometimes this strategy doesn't work. If so, try method 2.
Method 2: Take it to the dry cleaner.
That's what dry cleaners are for. They iron shirts. A simple wash and press costs less than a couple bucks. If you think about it, this is probably cheaper than method 1.
If (somehow) both methods one and two fail, proceed to method 3.
Method 3: Iron the shirt yourself.
This requires what's called an "iron". Preferably the iron lets you pour water in it, and even better it has one of those slippery teflon coated bottoms. Plug the iron in and if there is a dial, turn the dial to point to cotton or very-very-very hot. All Proper Cloth dress shirts are made from 100% cotton. Now, you need an ironing board. If you're in New York, you have a dinky little ironing board you can set on your dinky little dining room table. If you live in California, you're wondering what an ironing board is because you wear nothing but t-shirts. If you live anywhere else, you have a nice big fold out ironing board you can setup in your living room. Figure it out...
*Expert tip: If possible, position the ironing board so that you can see the television.
Step 1. Unbutton the shirt completely, and spread it out over the ironing board. Start with the back of the shirt because this is going to get wrinkled as soon as you put it on and sit down anyways. Put the dress shirt on the board so that the narrow end of the board is going into the back of the shoulder of the shirt and the edge of the ironing board is along the side of the shirt. With moderate pressure, slide the iron down the shirt top to bottom, being careful that you keep the shirt flat and don't actually iron wrinkles into the garment. Use a little steam or spray some water on any wrinkles that aren't going away easily. Once you've done one side of the back, slide the shirt over and do the other side just the same.
Step 2. Iron the sleeves (one at a time). Lay the sleeve long ways on the ironing board and carefully flatten the sleeve with your hands such that it folds along the hem on the bottom of the sleeve. Starting from around the arm pit area, iron toward the cuff and away from the bottom hem. You should iron in a crease on the top of the sleeve and this crease should be straight. Repeat on the other sleeve.
Step 3. Now we do the collar. Take out the removable collar stays from the shirt collar (don't lose them). Button down collars won't have collar stays, but you'll need to unbutton the little buttons that hold the collar down. Unfold the collar so that it is "popped" and lay it flat on the ironing board with the back facing up. Iron from the middle of the collar outwards all the way to the tips of the collar points. Because the collar is a thicker stiffer piece of material you may need to press slightly harder than you would for the rest of the shirt. When you're done, put the collar stays back in the collar and fold the collar down again.
Step 4. Finally, do the front of the shirt. Hopefully, by now you've got the hang of it. Do one side at a time. Be careful of the buttons. Make sure the front placket is not folded over in a way it shouldn't. Use the point of the iron to get in the areas up around the front of the collar. If dress shirt has a pocket, this can be tricky, but you can be trickier.
Still having trouble, go back to method 1. Otherwise, mission accomplished.