April 15 is the deadline, but you won't lose your green card if you miss it*. You'll just get penalized by the IRS if you owe taxes that haven't been paid by April 15.
You can apply for an
extension to file taxes, which is normally granted automatically and you'll get until October 15 to file it. However, it is not an extension of the time to pay taxes; you still have to estimate what you owe and send in the money before April 15. If when you file the actual tax return later in the year it turns out that your estimate was too low, you'll have to pay the difference plus a penalty. If the tax filing results in a lower tax liability than your estimate, you'll get a refund for the difference.
Note that as a green card holder you are taxed on worldwide income. There are tax treaties and credits that can be used to offset most or all of the double taxation if you are also taxed in a non-US country, but the rules are complicated and I would strongly advise you to use a tax professional (not a part-timer like people at H&R Block). There are firms with accountants who specialize in US+foreign country taxation. It may cost you a couple hundred dollars to use them, but if you try to do it on your own you either risk paying too much taxes because you don't know how to take advantage of all the foreign tax treaties and credits, or you pay too little and have to pay IRS penalties.
I went through a similar thing when I first moved to the US. I did my own estimate of taxes based on my incomplete knowledge and sent in an extension request in April with the money. Then a few months later I got a firm to handle the actual tax return. It turns out my April estimate was a little low, so I had to pay something like $12 in penalties. But if I didn't request the extension and didn't send in the estimated amount, the penalty would have probably been over a thousand dollars.
Note that if your worldwide income is under about US$8000 (after converting from foreign currency and including non-work income such as interest and dividends), you probably won't owe any US taxes and may not have to file. If you don't owe anything but you file taxes late without applying for an extension, there won't be a penalty, because the penalty is based on a percentage of what you owe.
*however, if you are outside the US for a long time (generally this means over 6 months straight, or multiple consecutive trips of under 6 months each but adding up to over a year), your green card may be revoked for abandonment of residence. When deciding to revoke it for abandonment of residence, they look at whether you filed taxes, among other factors. So you won't lose the green card purely for missing the deadline, but failing to file taxes on time (or at least apply for an extension before the deadline) could jeopardize your green card when considered in conjunction with your absences from the US.
