Installing new T8 ballast
This is what happens when you walk into Home Depot and buy a bunch of random parts.   Which is why I do not like Home Depot.   That, and their prices are outrageous.  

Currently, you have a genuine fluorescent fixture, with a rapid-start ballast.  (Or possibly the even better programmed-start type).   Hence, the 2 wires to each end of each tube.  The 2 wires use the little filaments in the tubes to preheat the tube, which makes it easier to start (strike the arc). A fluorescent makes a Tesla-coil style arc through a tube of argon gas. In fact Tesla invented this.  

What you bought was an instant-start ballast, which only has 1 wire per end, because it ignores the filaments and strikes the arc with a violent voltage surge.  This is hard on the tube, so don't use them where tubes are hard to replace.  I'd call it a cheapie ballast, but in fact it is not cheaper.   

You have not told us if this was originally a T8 or T12 ballast. (It is an option to stay with T12, and today's tubes are excellent due to new quality standards). I have been switching to T8 because my old T12 ballasts are failing, and I'm not buying new T12 ballasts. Nope.

You have no idea if the fluorescent tubes are bad, or if it's the ballast.  OK.  Are there black marks a couple inches from the ends of the tubes? 

Staying real fluorescent and switching to T8 is a good plan.

For that, you need to buy actual fluorescent tubes with the argon gas. Sylvania has some 90CRI tubes that I absolutely love, and I pay about $1.60 a tube in 36 quantity. 

You need to sort out whether you want to stay with rapid-start or the deluxe programmed-start ballasts; or switch to the poor instant-start type.  Instant-start does have one advantage, implied in its name.  It is not as likely to work well in very cold; for that go programmed-start.  All will not flicker.  

Your fixture's wiring can handle any type.  

To switch ballasts, if the old ballast is known-bad, cut the wires flush at the ballast.  Otherwise leave 6” so it could be reused. Don't tear too much wire out of your fixture, sourcing legal 18AWG 600V wire is hard, and sourcing the wrong wire is too easy.   

If you replace with a rapid or programmed start ballast, it's real simple - yellow to yellow (interchangeable), blue to blue, red to red, done.  Use blue sized wire nuts.   Don't bother fiddling with the sockets (lampholders/tombstones). 

If you replace with an instant-start, then one color (blue or red) has only one wire.  Both yellows go to that color.  Both blues from the fixture go to one of the remaining wires.  Both reds to the other.

Don't worry about the excess wire, just fold it back and put the cover over it, there's room in there.  It does no harm.

Now having ballasted it for actual fluorescent tubes, you could put LED replacement devices in there if you really want to, but ...facepalm... Why.  Modern real tubes are better light.  If you do this, you use LED types called Plug-n-Play, and you must make sure the subtype of plug-n-Play LED matches your ballast type - instant-start, or rapid/programmed-start.    But this is stupid, because you are forcing a ballast to drive something it was not designed to drive.   

Or use direct-wire LED replacement "tubes"

In these, you wire 120V straight to the LED "tube" and remove the ballast entirely.  These come in two types.  


Power on each end of the tube:  in this case you wire both yellows to supply neutral, and both blues and both reds to supply hot.  
Power on the same end of the tube: in this case you wire supply hot to one yellow, supply neutral to the other yellow, and cap off each red and blue wire separately. If you install the LEDs and they don't work, flip them around so you're plugging in the opposite end.