3.0 Soldering 101

If you've never soldered before:

(tips here)

Basics

Solder all of the surface-mount components first. This lets you lay your board flat on your work surface.
Start soldering the components from shortest to tallest. Once you have some components down, it'll be easier to manoeuvre your hands around short things as opposed to massive skyscrapers poking our of your board.
Adrianna also recommends beginning with the components on the inside of the board and working your way towards the edge so other components aren't in the way of your iron.
Always apply flux to the pad you're about to solder! Flux is magic, and more of it shouldn't hurt.
2-pin surface-mount components:
1. Always start by applying flux to the first pad, and touching your solder-covered tip to that pad. Let that pad get a nice amount of solder
2. With one hand, pick up the component with your tweezers. With your soldering-iron hand, melt the solder on the pad so it's nice and flow-y. At the same time, place the component on top of the footprint (making sure that it stays nice and flat on top of the board) and ensure that the pin makes a nice joint with the pad.
3. Apply flux to the other pad, then solder the other pin.
4. Give both joints a quick wipe with the tip to remove any excess solder.
5. This video shows the process:
7. (he does it slightly differently by placing the component on top of the un-melted pad, which works too, just make sure the component lays flat afterwards)
Multi-pin ICs:
1. Start the same way as above, by adding solder to a single pad. BUT this time, the first pad must be for a pin on the corner of the IC.
2. With your tweezer hand, bring the IC towards the footprint (MAKE SURE IT'S IN THE CORRECT ORIENTATION FIRST!!!!). As you bring the component closer, take care to ensure your IC isn't crooked relative to the pins and melt the solder on the first pad.
3. Move the component closer until the pin touches the pad. Hold the component steady with your other hand, making sure every pin is correctly lined up with its respective pad. When everything is good, take your iron off the pad whilst still tweezing the component to hold it still until the solder solidifies.
4. Next, solder the pin in the opposite corner of the IC. Now it's tacked down nice and can't move at all.
5. Solder the rest of the pins.

Adrianna's Advanced Techniques

Soldering with both hands

Yes, really. I taught myself to solder with both hands. My non-dominant hand is probably 70% as good as my dominant one. You're probably wondering, "why on earth would this ever be useful?" Consider this situation:

Let's assume I soldered the right pad with the iron in my right hand. To solder the left pad, I could turn the board 180 degrees so that the unsoldered pad is now on the right, and solder that with my right hand like normal. And this works most of the time. But I've found that when I'm soldering under a microscope, rotating the board throws me out of the loop because I then have to reposition the board under the lens, which takes time if there's loads of components on the PCB. I could also try to solder the left pad without a rotation, but that results in an awkward position for my right hand.

So I started switching the iron to my left hand and soldering like that. Try it!

The single-row SWISH

You've tacked down your IC with your initial 2 pins in opposite corners. Instead of soldering each pin individually in a row of pins, you can do this:

Put HELLA flux on the row.
Put HELLA solder on your iron tip.
Place your iron to the first pin and sweeeeeeeep the tip over the row, adding solder to all of pins. Some of the pins will be shorted. That's okay.
Touch up each pin by using your wick to remove any excess solder, as well as reflowing the joint to get that nice "sucked down" look.

This video is a good demonstration:

Sources:

https://learn.adafruit.com/adafruit-guide-excellent-soldering/