There are so many raids that drop mounts and any raid that isn’t current content typically has a 1% chance, or less, of dropping a mount. So you should settle in for many attempts and become accustomed to disappointment. This is my current, weekly farming route for raid mounts. The route is subject to change as I acquire said mounts; thus the potential for multiple parts.
There are a few addons and weakauras you can use for mount farming. Rarity seems to be the most popular mount/pet/toy farming addon. It’s highly recommended by my guild mates. I’ve just started using it myself so I cannot provide too much feedback, or support (at the time of this post).
You will also need your Hearthstone (which you should always have) and I would also keep your Dalaran Hearthstone handy. Also, make sure you have lots of space in your bags and a vendor mount is super handy (such as the Traveler’s Tundra Mammoth). Since some mounts only drop from specific difficulties I suggest to always have your Raid Difficulty set to 25-Man Heroic.