A lot of the expedient repairs I've done resemble the "dead bug" style, which would cause nightmares to any PCB designer (and myself sometimes), but if it works for getting you out of a sticky situation, then it's fine by me. If you have the time and resources, at least try to fix it properly later.
I got a chuckle out of this. I'm an embedded systems designer, and I've been designing circuits and PCBs since the 90's. On some complicated designs (and even some simple ones!), it's unlikely that the initial prototypes will be 100% correct. That often leads to some ugly and interesting rework.
Back to the OP, first, you need a place to work with decent lighting and maybe a static mat. For repairs, you need to be able to determine what's wrong. This typically requires tools like meters, scopes, and logic analyzers. If you travel or are just starting out, something like this http://store.digilentinc.com/analog-dis ... ro-bundle/
is good. I'd also get a small multimeter.
It's also very helpful to have schematics and PCB layout details for the thing you're repairing.
Other tools: I need a microscope these days. With good magnification, you can often find the source of a problem (poor soldering, cracked solder joints, bad etching on PCB traces, etc.)
Another very useful tool is a thermal camera. These can be used to find a bad component in some cases.
Also helpful is a lab power supply where you can adjust the output voltage and the current limit.
A good selection of spare wires: bare wire of all sizes on spools, various USB cables, test leads with alligator clips and hooks, power cords, etc.
A software defined radio can be used to "sniff" for signals on a PCB with a small, handheld loop antenna.
A signal generator is useful for injecting signals at various stages.
Resistor and capacitor decade boxes can be helpful too.
A good soldering iron, solder, solder wick, flux, picks, tweezers, wire, epoxy, prototyping board, and a good selection of spare parts help, too. The spare parts will depend on your skill level.
Finally, I think that development boards like Arduinos can be helpful. If you lose some logic chips, you can easily program logic functions into something like that.
These things take years to accumulate, so start with the basics and build from that. Ebay is your friend here!