I'd suggest you start with testing the resistors & diodes (and if you're getting the wrong values or no reading, try desoldering em and test em out of circuit for verification!) Also, do note that the values of the resistors are listed in the service manual given to you above.
Apart from that, also pay very close attention to the electrolyte capacitors (look for ones with a bulged top or leakage).
A healthy capacitor should always have a flat top. Do keep in mind however that while there are often physical indications of a bad capacitor, a capacitor can also look physically okay (but be faulty), so in order to fully rule out em all out, you'd need to have a capacitance meter (as your multimeter only supports up to 200 microfared, which isn't much) then you should either remove the capacitor from the circuit board, or desolder one of it's legs before you measure it (just add a bit of fresh solder to the joint, and use a wick, you don't really need to remove the leg from the board, just suck the solder out of it) but before testing any capacitor this way, you must first discharge it to be on the safe side (for low voltage capacitors, you could use a screwdriver or pliers to short both legs, for high voltage ones however, you need to either make your own capacitor discharge tool with the right resistor, wire and heatshrink, or buy a pre-made one) Personally, I use the UNI-T UT601 for testing caps, you can get it from here:
http://www.banggood.com/UNI-T-UT601-Professional-Capacitance-Meters-Ohmmeters-Capacitor-Resistor-with-Diode-and-hFE-Test-p-1043052.html
Or maybe order it from ebay if you want it delivered more quickly.
I also suggest you get special testing leads for it:
http://www.banggood.com/SMD-Inductor-Test-Meter-Clip-Probe-for-Resistor-Capacitor-Multimeter-p-912618.html
Raz, that's really not very helpful troubleshooting "method". Starting to check and desolder stuff at random before you even know where the fault is (Is it the power supply? The control board? Is anything obviously fried? Are the power rails correct?) is not helpful and will not lead to actually solving the problem.
Also, recommending a capacitance meter like that is pretty much useless. For checking power supply capacitors you need an ESR meter, not a capacitance meter. A cap could be dead as a dodo with high ESR and still measure in-spec capacitance - I have several ancient USSR made ones which are like that. As far as I can see, that cheap meter doesn't measure ESR but measures up to 20mF capacitance (yay, that's is really *super useful* - like those cheap RLC meters that measure inductance to 20H but start at mH range ...). Electrolytic caps have tolerances -20+80%, so measuring the capacitance of filter caps in power supplies is not going to do you much good - it is very rarely critical and in the worst case you get some increased hum somewhere. Increased ESR on the other hand will let significantly increased ripple in the circuit, which
will cause problems.
Moreover, on linear supplies the electrolytic caps are unlikely to be an issue unless the device is ancient, with the caps having dried out, but plenty of 20-30 years old pieces of kits have original elcos and no problem whatsoever - the capacitance is not critical, so even losing some due to aging is not a problem.
Buying that meter is a complete waste of money, IMO. Especially for repairing this stereo. And those tweezers leads ..

Did you actually check the photos? That boombox is almost completely through hole ...
To john8893:
You are lucky that there is a schematics - for diagnosing the power problem you will likely need only a multimeter to check the voltages against what is written in the schematics. No need to order anything special.
This portable stereo has a linear PSU, there isn't much there except a transformer, rectifier bridge and a few capacitors (left side of the first photo). Then there are several linear regulators for each of the voltage rails derived from the unregulated 12-14V or so that the PSU (or batteries!) produces. So one is unlikely to find leaky or bulged electrolytic caps there. I would say that the PSU is mostly alright, because standby works - and according to the schematics you posted, there is no separate "standby" rail, unlike on many switching PSUs. Basically the PSU is always on, even when the boombox is in standby, only the various loads (motors, amplifiers, etc.) are switched off. So that tells me the transformer, the diodes, the microcontroller actually handling the standby and its voltage regulator are all OK. The capacitors may be out of spec, but most likely they fine - again, it is a linear supply and the caps don't see any significant ripple currents at 100Hz (unlike in switching power supplies).
Measure the voltages from the PSU and all regulators, they are indicated on the schematics (p. 23, 24). If they are OK, then I would start checking the stuff that controls the standby line. If I read the schematics correctly, the power button is connected directly to the microcontroller using the PIN line. The MCU then controls the POUT line that arrives to page 24 again and controls the transistors Q914 and Q915 that actually turn the power to the amplifiers on. The power button is fed from the BUP line which is the backup voltage. That's on page 25 up, the transistor Q704. Normally I would expect that transistor to be conductive and thus the power button have one pin grounded. Check the voltage regulator feeding it (IC305), there are voltages indicated in the schematics you should see on the board.
If all that checks out, it is possible that the microcontroller is detecting some sort of a fault on power up and refuses to turn the main power on. You did check for loose connectors, broken wires and similar stuff, right? If that turns out nothing, then it will be tricky to find without knowing what the MCU actually checks to turn the power on

However, to be honest - unless the problem is something trivial like a busted regulator on the PSU board that can be found quickly by checking the voltage rails and comparing with the schematics, I wouldn't bother with troubleshooting this. The stereo is a very old piece of junk and most likely won't be economically repairable after you factor in the time you will have to spend on it. Especially if you discover that some oddball IC on the control board has failed - it may not even be possible to get a replacement for it in the first place. Unless your goal is mostly educational (learning to troubleshoot electronics), then buying a new stereo will likely be a more sensible option.