First song is [crossing field] by LISA, the first opening to Sword Art Online. I just marathoned all 25 episodes of the series in two days and plan on ordering the first volume of the light novel. I discovered the show thanks to an AMV shown at the contest at con, and would highly recommend it (though I personally enjoy the first half of the series more than the second).
One of my favorite openings for Fullmetal Alchemist Brotherhood is Again by YUI. I first learned about her from her song Rolling Star that was used as an opening for Bleach. She's currently still on hiatus, so there's been nothing new from her in over a year, but a good singer to check out.