What components are required to upgrade from a 14 speed to 24 speed?
First, make sure your bike is fit properly - with a bad fit, your efficiency is likely lousy. 

More sprockets is not necessarily going to make you go faster/easier - changing gear appropriately and becoming more physically fit will (along with better selected sprockets sizes - we went ages before the Gillette razor-blade increase in rear sprockets...). Most of the more sprocket drive trains are driven by hype. 

You may need better choices of sprockets, but certainly, with a 2 x 7 setup, you have enough choices that it should be fine (for example, if you were touring, a 11 cog in the rear would be useless - you'd want a range of larger cogs so you could move the stuff you're lugging around). A 24 speed would need a 3 x 8 setup which means new crankset, new front derailleur, possibly new bottom bracket, new cassette and new shifters (these are the real killer if you want brifters; bar ends or downtube or friction shifters somewhere are cheaper) assuming you could get the chainline and what not right (the RD will be fine if you stay shimano). At this point, with a cheap bike like the Mirage S (400 dollars), you're better off just replacing the whole bike, since parts and labor will cost more than the bike is worth/costs (you will also need some tools you might not have, like a crank puller and cassette tool and cable cutters and what not). 

If you're going for a sprint tri, you should be in decent physical shape, so I'd look at fit or just spend some time getting used to the gearing - whenever I haven't ridden my road bike in a while and I get back on, I feel like its geared way too high, but after a few weeks, I feel fine. 

If you want to change something and you're sticking to this bike, your best bet is to just change the cassette to something with a wider range of gearing. This may necessitate a new rear derailleur (go wide range (marketed with a SGS code) from Shimano). I guess a new chain may be needed as well if you're using weird gear combinations. I suggest you go to your LBS and see what they'd recommend for a new cassette if you want something easier, given the terrain you're likely to ride. 

TL;DR: Don't do this. Either get used to the gearing, put on a larger rear cassette or get a new bike. More speeds isn't necessarily better, knowing how to use them and picking the right speeds is far more valuable.