The Soldering Pen
The first thing you will need is a soldering pen. This is your magic wand with which you can cast electronick spells.
You must realize that the soldering pen is not just a simple tool like a screwdriver or a wrench. It is a brush that you use as an artist, it is a companion that walks with you through the forests of circuitry. The soldering pen is your friend.
Picking A Pen + Solder
Many folks will say you need this or that kind of pen. It all is really up to you what feels best. The main options are ones with variable heat settings or power ratings. I myself started out just using the single-setting Wal-Mart soldering pens. I never got one of those soldering "stations" - they are a bit much.
The pen I use now is the model in the picture above and I like it a lot and it costed about $15. I like that you can replace the tip easily. The only complaint I had was that the tiny knob actually fell off a few times and I had to just superglue it in place carefully. I have also had some problems with the power cord shorting out and had to resolder it back on a couple times but, if you actually take the pen apart it is very nicely designed actually, even if it is kind of cheap-y, it is very user-serviceable.
As far as solder goes, I use whatever really. Look into it yourself, there are a lot of options and everyone has different preferences. There is also a lot of debate around leaded or lead-free solder, the role of technical stuff like rosin and flux, etc. Just try out a few kinds and see what works best for you.
The more you use your pen, the more you will naturally develop a technique with it. Our brains are like this - we build habits slowly over a long time. Like riding a bike, you will be clumsy for a while and it will seem almost impossible, but this is just the way things go. If you've seen someone experienced working with stuff, you might notice their hands get into these weird positions and stuff - the famous "live long and prosper" gesture is a common one when you need to hold onto three things but only have two hands.
I will give some of my best tips below, I hope they help!
A Solder Joint Takes 1-3 Seconds
This is the most important thing honestly. While it might seem like a brag, being fast isn't just a fancy trick but is an important part of soldering! If you are lingering on a soldering joint for more than two seconds, something has gone seriously wrong. Either your pen isn't hot enough, you ordered the wrong type of solder (there aren't many different kinds but some kinds are very crappy), the tip is dirty, or some malevolent spirit is working against you.
It might seem too stressful to try and get a joint done in a couple of seconds, and you'll be tempted to linger for five or six or ten... But you have to realize, solder melts very quickly, and if it gets too hot it gets gunky and "old" and crusty and bad. Not only that, but most electronic components don't very much like being really really hot for more than a few seconds! It is a really hard thing to wrap your head around, and took me a long time, but you have to be patient and just keep trying. Just remember, being quick is important!
WARNING: Beware Tip Oxidization!
If your solder won't stick to your pen or components, and just balls up and falls off, it is likely your pen has oxidized. While you should be sure to keep your pen clean of any plastic or weird gunk, also be sure that you do not make it "too clean"!
Oxidization is the same thing that happens with rust, and it comes from over-exposure to oxygen. This can happen slowly over time, but is made worse by water and very high temperatures.
Always Tin Your Leads!
This might sound redundant, but keep in mind that the point of soldering is to connect two metal bits together so that electricity flows between them. Because of this, you have to realize that the solder (metalish stuff) has to actually fuse with the leads (metal parts for connecting) on both things! Tinning leads means to apply solder to them before connecting them.
You also should tin your soldering pen a bit as well from time to time to prevent oxidization - a very thin layer of solder can protect it pretty well.
Once you tin your leads, connecting two components becomes a lot easier because the only thing you need to do is melt the solder on the two leads together, which should only take a second or two if you used a good amount of solder when tinning. There are techniques you will learn to tin your leads well, so that they have a really nice clump of solder on them that you can work with when making the connection.
Keep Your Pen Clean!
Trust me, I didn't clean my pen at all for years, and this ended up meaning I had to buy new ones a couple times a year. While cheap $8 pens are fine, this is just wasteful! Keeping your pen clean is easy too - you don't need some fancy station for it. All you need is some copper wool. Copper wool is excellent, and you can get some pads of it for $5 or so. I have a copper wool pad handy and stab it with my pen like a pin cushion to get excess solder off my pen.
If you get some kind of gunk on your pen like plastic or rubber or something (you really shouldn't... but accidents happen), that is when you'll want to use a cotton rag or washcloth - but make sure that your rag is not very wet, because otherwise you might end up oxidizing your tip! It is always best to try and use some copper wool or even just scrape gunk off with a knife though - the less not-solder and not non-metal things that touch your pen, the better.
Even if you are not like me and getting all of your components and stuff from recycled junk scrapping off of boards, you will inevitably have to do this at some point. I heavily recommend buying a mechanical, hand-operated desoldering pump. it is just a spring loaded thingy which sucks hot solder off of a circuit board. they are tedious to use but worth it.