If you have never soldered before, please see the soldering is easy comic. It's a good read, even if you have soldered before.
Another good one is the Adafruit guide to excellent soldering.
These are the parts that come in the kit:
After soldering, the board should look like this:
The only components on the boards where the direction of installing them matters, are the 2 LEDs and the ATmega328p itself.
The silkscreen of the ATmega328p has a little notch cut out of one side. That corresponds with the notch on the chip and the IC socket. The pin closest to the white dot on the silkscreen is pin 1 of the chip.
The 2 LEDS should have their longest pin (+) in the hole closest to the edge of the board.
All other components can be soldered to the board without thinking about it. It doesn't matter if you rotate them 180 degrees, or the way you picked it up from the table.
Soldering the components
If you're looking for a step-by-step soldering guide for the AVRPi-328 kit, solder the components in the following order. Rule of thumb: lower profile components first.
3x 1K resistors. R1, R2, R3.
R1 and R2 are the current limiting resistors for the LEDs. The value 1K makes them bright enough to see them being on or off in daylight, but not so bright that they're unpleasant to look at and burn up unnecessary amounts of current.
Lower values make the LEDs brighter. Any value from 150/270 ohms or so, up to 1K are reasonable values. R2 is the resistor for the PWR LED. This LED is burning all the time, you'll probably don't want this one too bright.
R3 is connected to the RST button, and protects GPIO pin 8 on the Raspberry Pi in the (rare but possible) event that you press the button at the moment a 1 is written to the output pin on the Raspberry Pi, creating a temporary short. It probably won't happen a lot (if at all), but resistors are cheap. Especially when they can prevent damage.
1x LED green (PWR) Fwd ~2.4V / ~2500 mcd
The green LED is the PWR LED, The red LED is the USR LED.
Even though they're both limited by 1K resistors, the red LED is brighter than the green one because of the difference in forward voltage and luminous intensity between the two. The red USR LED is onandoffable (pin 13/PB5) and you can use SoftPWM to dim it.
These parts have polarity! Make sure to put the longest pins (positive) of both LEDs in the hole nearest to the edge of the board. There's a '+' sign near the right hole.
1x 10K resistor. R4.
R4 is a pullup resistor for the RESET pin. It makes sure the default state for the ATmega328p is "running". The value of this resistor should be 10K.
1x IC socket (28 pins)
These parts have polarity! The IC foot has a little notch on 1 side. Match that up with the notch on the silkscreen.
The ATmega328p has the same notch on it. There's also a little dot near the notch. The dot is also on the silkscreen, and is pin 1. They're a bit hard to see in the picture on the left, but they're at the bottom.
You might have to bend the feet of the ATmega328p a little bit on a flat surface (try a table) to be able to fit in the IC socket. Bend them as straight down as you can, but not so far that they point inwards. Wiggle a little and be careful when inserting the chip in the socket.
Install the ATmega328p after you have soldered everything else. It's in the way when soldering, and you don't want to damage it.
2x 0.1pF/100nF capacitors. C3, C4.
These 2 capacitors do some filtering and make sure the current on the chip is reasonably ripple-free and stable. They're connected to the VCC and AREF pins.
1x Tactile switch.
This is the RESET button. The chip is automatically put in reset and out of reset when you program the chip with avrdude, but sometimes you'll want to manually reset the chip. Or use 'avrpi -r' in the terminal.
1x crystal oscillator (8, 12, 16 MHz)
Solder the 2 brownish-red, disc-shaped capacitors on position C1 and C2.
It may make sense to solder the oscillator itself in after you've soldered the female pins (as the very last step). The female pins give a nice amount of space for the oscillator to rest on while soldering, so that you don't have to solder it tight to the board. Keep the pins straight up (don't bend) when you solder them.
The benefit of that is that if you ever want to install a different oscillator, it's easier to de-solder the old one. Its pins will be straight, and you can even cut it away with pliers before de-soldering without ruining the component.
7x various headers
Solder the 2x20 female pinheader on the bottom of the board. It plugs in the GPIO header of the Raspberry Pi.
The rest of the headers go on top. There's only one way to fit everything in their holes, so you really can't go wrong. Make sure everything points straight up.
Only solder the 2x3 ICSP pin header onto the board if you know what it does and have a use for it.
To decide which jumpers to put on the VIN and CONN male headers, see the CONN and VIN header on the pinout page for the AVRPi-328.