The tools for the electronics tradesman will vary between disciplines. The majority of work I do is amplifiers repairs, with some fabrication of enclosures and the like. An embedded design engineer would have tools specific to that area, but many of the tools listed here would still be relevant.
You will not need to purchase all of these tools at once, rather you will add each tool as it is needed over the course of an apprenticeship. A soldering station will likely be supplied by your employer. A nice pair of side cutters and a set of insulated screwdrivers are essential. As soon as you can afford it, purchase a large rolling tool chest with sliding drawers.
As you accumulate more tools, careful attention should be paid to organisation. It saves a lot of time if tools used for particular tasks are grouped together.
Toolbox Top: large frequently used tools.
Probes + Clip leads for multimeters.
Two Multimeters; Measure voltage and current at the same time or monitor dual supply rails.
Discharge resistor 25W 40R, used to discharge supply capacitors before working on a PCB.
Helping hands for holding connectors when terminating.
SMD beak; a small pointed arm that holds SMDs in place while soldering.
Small socket set, Hacksaw and safety glasses.
Stanley security driver set.
IR thermometer; check heatsink thermal protect circuits are working correctly.
Small top drawers: Small tools.
Combination spanner set.
Punches, deburring tool.
Small screwdriver set.
IDC removal tool, Tweezers, Prying tools.
Torch, Otoscope for inspecting small devices.Markers.
Second drawer: Most frequently used tools.
Insulated screwdriver set.
Bullnose pliers, Wire strippers, Long nose pliers.
Screw removal pliers.
Small long nose pliers.
Side cutters *2; keep an old pair for cutting steel wire.
Third drawer: Measuring tools.
Square and 12" Ruler.
Screwdriver extension bit.
Microphone for testing audio inputs.
Fourth drawer: Fabrication tools.
Files + wire brush or file card.
Ball peen hammer.
Fifth drawer: Infrequently used tools.
Large socket set.
Stanley knife blades.
Above workbench: Test equipment
Curve tracer for power-off in-circuit testing.
Mains current limiter.
Speaker impedance tester.
Dual rail lab supply.
Device programmers, Connectors, Test CDs.
Scope probes and Differential Scope Probes
Solder, solder wick, solder sucker.
Containers for holding screws.
A good quality temperature controlled soldering iron is essential. I use the JBC CD2BB, as it has easily removable tips and heats up very quickly.
A PC modified with a power button on the rear of the case allows easy access to all ports.
I also find it necessary to wear a tool belt, so that I always have these items on hand:
Markers and pen.
Small note pad.
Tape measure (for measuring freight boxes).
Folding stanley knife.
A lot can be learned from tools of other trades; Tom from OX Tools produced a great video on tools for the apprentice machinist.