We wish to set up our working lab for the Arduino and seeking some advice on purchasing some sort of electronic tool kit, like the one in the pic attached. For those who have a well established lab that already have all the tools to work on Arduino projects, it would be great to get a good start and then we can add to it.
Buy an Arduino, a multimeter, a soldering iron with solder, some side-cutters and some plug-in breadboards for starters.
Also buy some single-core wires for the breadboards, and maybe wire strippers, although they're not essential.
Some heatshrink tubing in a few sizes can be pretty handy, too.
Much of the stuff in your pic would be a waste of money. There's not much call for ball-peen hammers with an Arduino, or oil, circlip pliers, spanners etc.
Rather than buy a lot of crap that you'll never use, buy the basics, then add tools as you need them.
Edit: More important than tools, beyond the basics, are some LEDs, pushbutton switches, some NPN and PNP small-signal transistors, maybe a few N-channel MOSFETs, a range of resistors and some capacitors in various types/sizes.
A good temperature controlled soldering iron is essential. Always buy good quality tools, they pay for themselves with reliable use.
Side cutters, pointy nosed pliers, posidrive screwdrivers (most electronics use these rather than Phillips head), flat blade screwdrivers, set of jeweller's screwdrivers, magnifying lamp etc.
I also tend to agree that you buy tools as you need them as there is a lot of cost for a set when you may not use many of them.
Soldering iron - Don't get a new, crappy soldering iron; I recommend a used weller, the kind with the teal blue case (some of these are electronically controlled, some are magnetically controlled - but either works, and the magnetically controlled ones will outlive you)
Philips, posidrive, and flat head screwdrivers. Set of jewlers screwdrivers. If at all possible, go with real screwdrivers, not the ones where you get one screwdriver, and a bunch of bits - the bulky head that you put the bit into keeps you from getting at some screws, and they're just less pleasant to use.