Whether it’s a horse, a trained wolf, or a griffon, from time to time adventurers find themselves fighting while riding another creature, also known as “mounted combat”. Fighting atop a steed, noble or otherwise, can be a great option. A mount typically offers better speed, and in some cases can offer additional movement options like flight without casting a spell.
Unfortunately, the published rules for mounted combat aren’t great. The basics work (mounting and dismounting, movement, the rules for creature size, etc.), but there are a lot of simple unanswered questions like how melee attacks and reach work while mounted. I wrote a Practical Guide to Mounted Combat a while ago which goes into detail on how the rules work and tries to answer some nuanced questions about how they work.
If you’re not ready for something that complicated, I suggest a guideline to keep things simple: no mounts larger than medium size in combat. That means mastiffs and ponies are likely your only options, unfortunately, but medium mounts remove essentially every edge case in the rules. Your DM will need to play along or find a reason for horses and other mounts to conveniently vanish whenever a fight breaks out.
With all of that in mind, if you’re still interested in mounted combat, see page 198 of the Player’s Handbook. Any meaningful addition I could make to the rules text is presented in my Practical Guide to Mounted Combat, and I don’t think it is important enough that you need to understand it when you’re first learning to play.