Soldering
If you come from a background, like computer science in my case, that doesn't involve much hacking of the physical world, beyond occassionally plugging things in, then soldering can seem a little intimidating initially.
But it's actually very easy and just takes a small amount of practice to get started. You do however need some equipment first.
Something like either the beginner toolkit (US$30) from Sparkfun or the similar but more expensive electronics toolkit (US$100) from Adafruit.
I'd be more inclined towards the Adafruit kit (even though it includes some things, like a breadboard, that you don't need). The Sparkfun parts, like the soldering iron, are really very basic (hence the low price). And the Adafruit kit includes leaded solder while the Sparkfun one includes lead free (see later).
Both kits include soldering irons suitable only for 120V countries like the US. For 230V countries, like the UK or Germany, you could combine such irons with a voltage converter - like this one for the UK or this one for Germany (the review ratings for both vary wildly).
Or you could just replace the iron with something like this Antex XS25 for the UK or like this Ersa 30S for Germany. The Antex has a silicone mains cable, there's a cheaper version with a PVC cable but the increased flexibility of silicone is nicer. For something a bit nicer but still in the hobbyist price range the Hakko FX-888D and the Weller WHS 40 are possible choices.
If you've got a soldering iron you're going to need solder. The solder, used for most electronics purposes, used to always contain about 40% lead. But in 2003 the RoHS directive was adopted by the EU and most electronics now use RoHS compliant, i.e. lead-free, solder. Hobbyists however can still legally buy and use classic leaded solder. Some people find lead-free solder fine to work with, or say you get used to it, but a lot of people (myself included) find it much harder to work with.
So for beginning soldering I recommend leaded solder - something like 60/40 rosin core solder. For a nice explanation of the main terms relating to solder, such as flux core etc., see this Electronics Stack Exchange answer.
While lead can be highly toxic there seems to be little risk associated with its use in solder, and little evidence that the lead later leaches out when items end up in landfill. The fumes that you see when using solder are from the flux - and not lead related - that said they're still not very good for you so make sure to solder in a well ventilated room.
For a nice getting-started project I'd suggest one of the Sparkfun learn to solder kits like Simon Says or WeevilEye.
Once you've got all the equipment, soldering iron and so on, and a nice kit like Simon Says it's time to learn how to actually solder. There are no end of videos and tutorials on the web but I suggest the Adafruit one and the Sparkfun one. Both include excellent overviews of all the equipment etc. before getting onto actually soldering.
Note: everyone seems to solder in a slightly different way so don't be surprised if the advice isn't quite identical across videos etc. but on the whole the core points are always the same.
Painless360 has produced a lot of nice RC related videos that are referenced elsewhere here - and he has a nice "Introduction to soldering" video. Similarly Propwashed have a "Soldering Exercises for Drone Building" video.
Update: I found Bruce Simpson's soldering basics video later and think it's excellent (though he could have sped up the bits where he's measuring temperature). He covers RC relevant aspects of soldering that I wasn't aware of from having just soldered up small electronics projects such as losing too much heat when using a low power soldering iron and the consequences of burning off all the flux in your solder.