I guess I will attempt to provide my knowledge and experience to aid you in your question.
How to get into competitive battling you say? Well you don't actually need your DS for that.
Pokemon Showdown is a very popular battle simulator that allows you to make your dream team and battle in your chosen tier. When you have built a satisfying team you can continue to play on Pokemon Showdown (free of charge) or you may want to build it in game and battle over wifi.
I guess next you need to learn about tiers. It's important that you pick a tier and stick mostly to the Pokemon inside of it. Tiers are fan made groups for Pokemon based on how well they preform inside all tiers. Take Dialga for an example. Dialga, like any legendary has monster stats and a scary movepool. Because of this you can battle with Dialga in the Ubers tier which is the top tier, usually consisting of legendary Pokemon. The next tier down would be OU in which strong Pokemon such as Garchomp and Scizor lie in. Uber Pokemon may not be used in OU and OU Pokemon are not allowed in UU, the next tier down. However UU Pokemon are perfectly OK in OU and Ubers as is OU Pokemon perfectly fine to use in Ubers. This is so that when you make a team, you battle opponents with Pokemon of similar strengths to your own making it a fair match. Each tier has different rules you must stick to when building the team. Here are Smogon's B/W tiers. They have not released any X/Y tiers yet but you can expect most B/W OU Pokemon to stay OU. If I were you, I would start off in OU because you have a wide range of Pokemon to try out.
There are many factors that go into building a good team. These are learnt through experience however in order to build a good team, you need to play your Pokemon to their advantage. This means having a wide range of coverage and having your weaknesses covered between your Pokemon. A team of pure Water types won't work because you have no Pokemon to cover for their electric weakness. Learning what works and what does not comes through experience. By battling opponents, you will pick up common Pokemon cores and strategies.
The next and last thing I will cover being the only other essential part of competitive battling is the battling aspect. You are no longer playing the stupid AI players in game but real people who have brains. This means making the obvious play wont always work out. They wont keep their ferrothorn in on your Delphox. They may switch out into a water type to take the hit in which case you need to predict. You need to learn basic prediction skills and how to read a player. Remember that they will be predicting you also.
I would reccomend you get familiar with the mechanics of Pokemon battling and read these two articles that will really help you improve as a battler:
The Art of Psychological Warfare
Risk/Reward <- very important.
To improve on your skills as a battler you can learn quickly from the "pros" or any experienced battlers. I shall reccomend some Youtubers to you who record their battles and post them running you through their thoughts as they make each play. You will find it incredibly helpful and they are actually quite entertaining.
Not Blame Truth
All three Youtubers are very good battlers and upload regularly. I know Youtube has helped me a lot and I am now a pretty decent battler.
Finally I would recommend practicing. You may lose once or twice. Hell, you may even get thrashed 6-0 but we have all been there before and giving it up is not the way to improve.
You can be cheap, almost guarenteeing a win and use this monster
Hopes this somewhat helps ;)