'Starting Competitive Battling'... well I'm not completely sure how to answer this :P
You get on Showdown/Pokemon Online and just press random battle! XD And then you're probably into a rated battle where you will just spam attacks :3 and probably lose
First off, you could try getting a tutor here, or just asking around in Chat for an experienced player to help you out into Competitive battling. This is the link for that on DB.
Just make your tutor do all the work and teach you like his your personal slave <3
Anyway, if you plan on getting into it yourself, for first getting some basic knowledge on it, you'll need to understand the concepts of Competitive Battling. Those being, the different Tiers (PO and PS both have seperate tiers, so if you decide to play on both simulators there will be some differences), EVs, IVs, STAB, Natures Team Synergy, Prediction and that shizz :3. You can find most of that on this website (EVs/IVs here), though things like Prediction are a bit harder to learn, and Synergy can vary.
When choosing your team, it should work together, and not all be killed by a singular Pokemon/type (That's synergy bla bla bla). Smogon University can help you with sets/evs and basic battling and all the stuff (they also have the tier listings for each Pokemon, which is the one Pokemon Showdown uses as well). They also have example teams (most being pretty basic), so you can use those first if you want before starting to make your own.
Pokemon Showdown Link - you can play Online or Download this one
Pokemon Online Link - You have to download this one to play, but I like it better c: probably since I used it first
I'll put more later c: