The first 2 heroic bosses I downed were moorabi and sladran from gundrak - i've got a dk, 3 spriests and pally healer, so moorabi is actually the easiest one for me, i just let him go ahead and change, the pally can heal the tank all day and the aoe thing i just holy nova x3 and go back to burning.
Sladran was harder for me until i concentrated on gearing more dps, now i get him about 90% of the time.
Doing Eck was next, he's really simple if you just max range tank him (35+ yards) and he won't jump. I had to glyph mind flay to get him.

Next i went over and spanked the root shaper boss in nexus - just clump up right behind the tank on the steps and it's simple.

Also, while there, I go up and get the first boss in the oculus, the trash is harder IMO than the boss. just run up with the tank on this one too, ignore the bombs and just do a little aoe healing as needed (i made it the first try on just VE healing).

And last on my current list of starter bosses is prince from UK, just make sure to have the /target frost tomb part in your dps macro.

These are the ones i'm farming for badges right now, getting about 6/day until i can finish a complete heroic myself.

I get killed on the collosus in gundrak as soon as he transforms (i have no nature damage mitigation, and my priest holy nova healing works best when they are all clumpped up, which can't be done on this boss.

I almost have the twins in UK down, just need practice.