I'm no expert at electronics but I have put a LOT of electronics kits together. A few suggestions:
Get a good soldering iron and practice with it a bit. This will pay off many times over. DON'T USE A SOLDERING GUN! It's much too hot for this and can damage your components. Make sure you tin the tip of the iron before you use it the first time.
I'd suggest getting a "Third Hand" tool. Here is an
example of one. This isn't required but I find it invaluable in making clean connections. I generally have one clip holding the board and one clip holding the component. Then I hold the solder in one hand and the iron in the other. Without the Third Hand, you end up setting the board down and hoping it will balance on the component and hold it in place. This never works out well.
I'd also suggest a "solder sucker". Here is an
example. This way, you can easily remove a glob of solder that you screwed up on. It WILL happen and a solder sucker can pull the molten solder right off the part and leave things ready for another attempt. A spin-off of this is "solder wick". I don't like this nearly as much as you have to really heat things up to draw a lot of solder off but it works too.
Clean the circuit board with a light abrasive before you begin. Something like steel wool works wonders. You want the contact pads to be nice and shiney. That will make your soldering much easier. Be gentle though as the pads can seperate from their foundation on a poorly made board.
Bend the leads of your components with a pair of pliars before you place them in the board. Not only will this look much better, it keeps you from forcing components too much. Certain components (like ceramic caps) are easy to break and a beast to replace if you don't know exactly what it was to begin with. Many of them are very poorly labeled if at all.
Take your time, go slowly. Triple check that the component you have selected IS the right one. Many of the components like diodes, ICs, electrolytic capacitors, etc. have a polarity or orientation that must be correct. Be mindful of it when you place the parts. When you do solder something, try and heat the pad and the lead at the same time. Apply the solder to the work, NOT the soldering iron tip. A well soldered connection is bright and smooth. Solder will "flow" around the joint when things are heated properly.
Personally, I like to start in one corner of a board and work to the opposite corner. This seems to give me the most room for component placement and just makes things easier in general.
Your board has nice large traces so this shouldn't happen much but if you accidentally bridge traces, you can use an x-acto knife to cut the solder between them in a pinch.
As solder gathers on your iron's tip, you need to clean it off. An easy solution is to have a folded square of damp paper towels. You can just rub the iron tip across this a few times and the crap will come off and leave the tip clean.
Don't let things get too hot. Almost all electronic components are heat sensitive to varying degrees. If things start heating up, stop for a minute and let them cool down. This goes for you too

ICs are the most suseptible, but other components might be as well. It's best to play it safe and just keep things as cool as possible while still getting good joints. This often means that I have to stop a joint and redo it once it's cooled down. It sucks, but this prevents you from buying another kit if you screw up a hard-to-replace part.
One last thing, WEAR SAFETY GOGGLES! Everyone assumes you can't get hurt with solder, but if it splatters in your eye, you can be blinded. I'm on another mailing list where a guy had a component explode and the solder nailed him on the upper cheek. A few millimeters higher and he would have lost sight in that eye permamently. Safety goggles are a small price to pay for that piece of mind.
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Whew, I didn't realize I had written so much, sorry about that. I hope this is useful to you. Good luck!
Mike