On my Omega Ruby game, I chose to get Torchic as my starter which eventually I evolved into Blaziken. Then, using my blazikenite I can mega-evolve Blaziken into Mega-Blaziken, giving him the ability Speed Boost. I know that there are moves such as Electro Ball that get stronger when the user has higher speed, but is there any moves that my Blaziken can use that would be able to take advantage of his speed stat gains?
Their aren't any relevant moves that Blaziken can use that specifically give bonuses based on Speed. The best move to "take advantage" of Speed Boost is actually Protect, giving Blaziken 'free' turns of Speed Boost buffs by being (almost) invulnerable to damage (the move 'Feint' hits through Protect).
Following on from that, generally the best way to take adantage of Speed Boosted Mega Blaziken is to outspeed & hit as hard as possible. With the combination of it's high attack stats, coupled with access to powerful moves like Flare Blitz, Overheat, Superpower, Knock Off and Stone Edge, Blaziken can be a real threat for your opponent to contend with.
For a more in-depth look at how Mega Blaziken works in a competitive sense (in both single and double battles), you should check out the guides on Smogon which can give you a broad idea of how to run it which you can then tailor to the needs of your team.