How to know if casting directors prefer native accents or a put-on accent
Sometimes breakdowns will ask for native accents only, which means at this stage they only want applications from people for whom that accent is their natural one. Accents to a native standard means people from anywhere who can speak the accent to the same standard as a local.
It's more likely that you will be asked to do an accent for stage work rather than screen work, but you may still be asked to do accents for both, so it's definitely still worth learning as many accents as possible to as high a standard as possible.
When it comes to a casting, if you have been asked to audition then you stand as much of a chance as someone who has the accent naturally, otherwise they wouldn't have called you in so don't worry about that. As long as your accent is up to scratch, it won't be an issue any more, they will be concentrating on how well you fit the character or how well you match the person who has already been cast as your brother/mother/boyfriend etc.
When it comes to learning new accents, think about which ones will be most useful to you specifically, and which accents are most commonly used in films and plays. I would say all actors should try and master RP, a solid general American, Cockney/London and either Yorkshire or Lancashire. That gives you a decent coverage of the most common accents and you can add to them once you have them nailed. But do think about your casting as well. The industry works in stereotypes, so a Cockney accent might not be as useful to you if you usually play lawyers and doctors for example.
You can add other accents like Irish, Scottish, Welsh afterwards which are all useful. Although saying that, it's more common to see breakdowns requesting native speakers when it comes to these accents. Also, more specific American accents like New York and the Southern bumpkin accent can be useful to have in your armoury.
When you are learning them make sure you are not doing a caricature; authenticity is key. Not many Irish people sound like spritely leprechauns for example. Talk to friends who have the accent you are learning, use online video streaming websites (not sure if I'm allowed to name them here) and really listen to the sounds they use, buy the accent books and CD's if you want, but I have always found it easier to learn by ear than read about it. Find a short passage to use until it sounds accurate, and hopefully the next passage you try the accent on will sound authentic much faster. There's a whole bunch of free stuff only a google search away.