Well we have to sort of leave the mindset that some people have accents and others don't. Everyone has their regional variation that they grew up hearing as well as a cultural/ethnic background from the home that influences how they speak.
How much their personal variation is different from what is accepted as that country's "standard" vocabulary set and phonological pattern is what makes a person "have an accent"
This is all really obvious so far but the strange part is why do we have a "standard"? What or who makes that the "correct" way to talk? As my favorite linguist once said: "A language is just a dialect with an army and a navy" (McWhorter). In other words, it's mearly socio-political power that gives a particular way of speaking legitimacy.
If the capital of the US just happened to have been founded in TX where you live, then it's most likely that your accent would be the "way everyone should talk" and my Buffalo accent would make me sound like a real redneck.
As to how to drop particular ways of speaking, there are a couple of ways to get there. The lucky people are those who are really influenced by the way others talk and can quickly pick up on jargon, vocal rythym, tone, customs, pronunciation and other verbal and non verbal cues. The rest of us have to study all that and use our exposure to these different ways of communicating to practice it really hard.
Most people in the world have multiple languages or dialects they learn or practice over the course of their lives with varying degrees of success based on a combination of factors such as natural talent, necessity, and passion or determination to know how to communicate in these ways.
The reason why most English speakers only know English is the necessity factor is very low. We don't need to learn another dialect/language to play, watch tv, go to school, buy and sell etc. Some people have to learn one for each of these things.
You don't have to worry about accent when you speak a very universal language.
Class dismissed for the day.
Stay awesome.