There are several things that could cause a door opener to not work properly. Tester101 put most of them in his comment, but most of those would at least result in the opener visibly and audibly trying to open the door, and then giving up.
This may sound counterintuitive but my first priority would be getting the light working. Most garage door openers use the light as the primary indicator of a problem, by blinking it a certain number of times. Replace all bulbs in the unit (preferably with bulbs designed for use in high-vibration applications) and see if the light will turn on.
The second thing to do is verify that whatever control you are using to try to open the door is properly connected and/or programmed. Replace any batteries, check the inputs on any hard-wired switch or panel, and have the opener "re-learn" any wireless control modules you're using. Understand that many systems require at least one hard-wired multi-function switch in order to access all the features of the opener.
Also check all the safety interlocks. Most openers on the market will not operate at all without the electronic eye system properly installed, connected, aligned and the lenses cleaned. Certain systems may have eyes at multiple levels to allow the eye to "see" things that could cross the threshold at different levels, like a car bumper vs a truck bumper; every installed eye system, and there must be at least one, must be working properly.
Next, pull the red handle on the traveller shuttle to disconnect the door from the drive mechanism. Press the garage door button, and see if the chain drive operates. If so, the problem is the door; it's locked, jammed or improperly counterbalanced. If it still doesn't move, the problem is the drive mechanism; either the travel settings are messed up, such that the door thinks it's fully up or down no matter what position it's really in, or else the motor has burned out, seized up, slipped a gear, etc. Try adjusting the travel and see if the drive will move. Also, some openers are pressure-sensitive and will abort closing if they sense that the door is pressing on something (like a kid who tried to slip through and got caught); that pressure sensitivity is often adjustable, and you may have it set to be too sensitive so it's tripping based on the weight of the door itself.
If none of these at least indicates a further course of action, you're probably looking at replacing the door opener.
I fixed the exact same problem with my garage door using a general purpose relay output wireless transmitter / receiver module I purchased from eBay. The receiver module for inside the garage operates from a +12VDC power supply which I purchased for low cost from Goodwill. The normally open relay contact of the receiver is wired in across the same wire pair used by the inside door open push button. The existing push button is designed to short across the two wires that run to the garage door opener. The added relay connections need to be wired in parallel to the switch so that they short across the same wires when the remote is activated.
The trigger buttons are small key fob type push button controls that operate from a standard type of coin battery.
Note that models on eBay change often and it is a good idea to search for a unit that comes with multiple fobs and to purchase a spare unit if you ever think you will need replacements. Some sellers will add additional fobs to your order for an additional small charge. The search to use on eBay is Wireless Remote Control.
The whole setup cost me about a third of the lower priced options that you have linked to. The unit I purchased was a two channel device with an A and B button on the fob and so one setup was deployed to run both of my garage door openers.
These come with the remote transmit code set to some default value and it is advisable to change that code to some random code that you select. Changing the code entails some work with a small electronics soldering iron inside both the receiver and the transmitter.
Best Answer
I'd start by opening up the opener, and making sure the wires for the switch are securely connected.
Inspecting the cable for damage, would be the next step. This would include testing the wires for continuity, to make sure there's not damage that couldn't be seen.
Check the control board in the opener for any fuses, and make sure any fuses are good.