When faced with computers that won't turn on, I find I only have a 50% chance to fix it - that always depends on what exactly the cause is.
First, the unreliable step - with the case open (to look inside), turn the power off from the wall, then turn it back on. Immediately after, press the on switch. From experience, this'll create a power surge that'll fix it 10% of the time. If it doesn't work, the important thing to check for is if the fans move at all.
If the fans don't move, double check the front panel headers, specifically PWR_SW. You can just try connecting the pins with a copper wire or something for a second to try triggering the power up sequence, although it's exceedingly rare for something to go wrong with the headers outside a broken switch (from pushing it too hard) or static short circuit (reseating the FP connectors will fix that).
If the fans do move, then it's a little more complicated. The green light can either be from an ethernet port (unusual to be lit when it's off, except with certain BIOS configurations), or from a "live indicator" on the motherboard (to tell you it's got power going through it and you shouldn't poke it too much). If the fans move but only get less than 10 revolutions, then it usually means there's a problem with the power phasing of the motherboard - either a short circuit or a failure to get enough power on the 12V rail (the big power plug that plugs into the motherboard).
I have mixed success in fixing computers at this stage of diagnostics. I usually unplug everything from the motherboard (bar the CPU, although I do reseat the CPU fan connector), check all the capacitors for bulging caps, give it a good clean to remove dust or hairs (hairs are common causes for short circuits), check all the connectors for dirt, then start putting back the important parts (graphics, 1 stick of RAM, no disk drives). It's often advised to rub gold contacts (for the graphics and RAM) with rubber to clear any static from them, although I tend to skip that step). Then I try powering it on again (with only the minimum hardware to see if it works). Depending on the symptoms after that step (e.g. if the fans spin slightly longer than before), I try other stuff (that's a bit too vague to list in the forums).
The most reliable way to test things is to have a spare computer to trial each of the parts (power supply, graphics, RAM, disk drives) and narrow down the cause of the problem. The PSU is the most obvious place to look first (a weak 12V rail can cause all kinds of hell in a computer), whilst the Motherboard is the most common part to fail whilst displaying this kind of symptom. If the motherboard's power phasing fails, replacement is the only option. If it's just a short circuit, then grounding the motherbaord (to clear the static electromagnetic field) and cleaning it should fix it. If it's the CPU, it's more annoying to determine (you'd have to try putting a different CPU in to test - I don't know any other way), so you might aswell just replace the motherboard and maybe get a new CPU to go with it.
Final comments I'll make - computers have a life expectancy relative to the quality of parts. Budget motherboards tend to last 8 months to 2 years. Budget power supplies tend to have 12V rails that deteriorate over time and will probably explode if it takes any more than 70% load. Budget cases tend to be bad at cooling the system and reduces the lifespan of every component by half (or more in the rare case of a high-powered system being put in a budget case). Non-Seagate HDDs tend to last 2-8 years (usually at least 5 years). No guarantees for any Seagate HDD - I don't know any that still works for me, my friends or my clients (the people who pay me to fix their computers).
