Where can I find Majors and Ultras for each species?

The best way to do those "kill 10 majors" bounties is by going to the appropriate planet. Earth for Fallen, Moon for Hive, Venus for Vex, and mars for Cabal. Do missions on hard difficulty to get nearly all majors in one go.

For example, the Moon's Sword of Crota mission on hard has 10+ Hive Majors in it, and it's a very quick mission.

A Major/Ultra is any creature with a yellow health bar.

EDIT: Since a recent patch, the Sword of Crota mission now only has about 5 majors in it, but that is still better than wandering around in patrol I guess.

For the fallen, do the "Scourge of Winter" mission; the Kell-Guard or whatever they're called count as ultras, and you fight a decent number early on.

I've found that doing the strikes, especially on heroic if you're high enough level, nets you Majors pretty quickly since they're the sub-bosses in each area. And the end boss is an Ultra with usually a Major or two as well.

For Fallen majors, go to the entrance to the Winter Kell on Venus. The Captain there is always a major. And the respawn on that group is six seconds. It will always be a major. This was where I farmed for 200 Fallen headshots, too. The whole thing took me 15ish minutes to obtain. Just stay on the upper level, and then move back and forth from on top of the bigger group to the cave entrance.

For Hive majors, load up the patrol for Earth. Go in the first building on your left. Hidden in the bottom is three Hive majors (two acolytes and one knight). All three will always spawn as majors, and the respawn time is also six seconds. Any time I have the hive bounty, I am done in a couple minutes.

I have rarely spent time hunting the Vex, so I defer to other information.

The Cabal spot for me is the same listed above, but with a twist. I go into the dome, wipe out the 3-4 enemies, and then come outside to wipe out the respawning 3 Cabal foes just in front. Each groups spawn in six seconds. I was able to grind up 200 tears in minutes there.