1) To max out your reputation level you will first have to buy the Stormwind tabard from Captain Lancy Revshon and then run dungeons repeatedly while wearing the tabard. However, if you have a human character (class trials included), you can skip to step 2.
With the tabard equipped any dungeon can be used for the purpose of the rep grind, but generally the better dungeons to run are Botanica Normal or Magister’s Terrace Normal. Thus, this grind can be completed entirely passively and can be combined with any other instance farming to maximize the efficiency.
2) After reaching exalted status with the Stormwind faction the mount can be bought for a tiny amount of gold from a vendor Katie Stokx. Humans can buy their faction mounts regardless of their reputation status. Thus, the fastest way of obtaining the mount is creating a Class Trial Human character of any Class, sending them around 50g and using it to buy and learn the various Wolf mounts. The class trial character can be deleted afterwards.
Extra Tips: Humans receive 10% extra reputation from all sources due to their racial buff. Additionally, countless reputation buffs can be utilized to speed up your reputation gains further, such as the Darkmoon Faire Buff (DMF is around for one week every month), WOW Anniversary buff or other mini-holiday related buffs such as Hollow’s End one.