The quest to unlock your Diablo 4 mount isn't available right at the beginning of the story. In fact, you'll need to work your way through to the end of Act 3 of the main campaign to access it. Even if you've unlocked the mount trophy during the server slam weekend, you can only equip it on your trusty steed once you've unlocked your first mount.
The world of Sanctuary is large, but you'll have to spend the first 30 levels running around on foot. At least you won't be tempted to dash past enemies that can give you valuable XP, and you can fast travel around the world using waypoints when running is just too strenuous. That said, if you want to learn exactly when you can expect to unlock the Diablo 4 mount quest, here's what you need to know.
Diablo 4 mount quest unlock
Once you've completed Act 3 of the campaign, you're sent to Kyovashad to speak with Donan, who you can find in the Cathedral of Light. If you stick with the campaign and don't do any side quests, you will likely reach this point at around level 30. Once you speak with Donan, you'll receive the quest Mount: Donan's Favor quest.
Find Oskar the Stable Master, just to the southwest of Kyovashad's waypoint. You'll get your first mount for free, though you can buy two others from the vendor section of any stable master's menu for 20,000 gold each. Each mount you own is automatically unlocked for every character on your account, so you won't have to buy them again. You can also customise your mount with trophies and barding at any stable master.
To mount up, simply call your mount with the Z key. You can get a burst of speed with your evade key while riding. One thing to be aware of is that your mount will get scared and spook if they're around too many enemies—if the bar above the horse UI head fills completely, you'll be knocked off.
Each class also gets their own "mount skill", which you can use to dismount quickly and jump straight into a fight. The sorcerer skill, for example, throws a wave of ice at the spot where you dismount, briefly freezing enemies and allowing you to get a few hits in before they can respond.
At the time of writing, you can buy only two additional mounts, and all stable master vendors seem to stock the same ones. Presumably, others may become available at higher levels or be tied to challenges or quest rewards.