Becoming a Friend

The "ally quests" are those that strictly reward follower XP. You'll only see the blue "!" if you have your follower for the day already summoned.

There should be one such quest at the bulletin board at Newhome, and (typically) two more out in Nazjatar -- things like jumping onto geysers. However, at the time you get this Becoming a Friend quest, those particular ones may not be available yet; you might have to unlock a few more plot points first. Other ally quests are available, and if you hover the mouse over the blue "!" on the zone map, they will be labeled as ally quests.

don't leave the town without your cohort in tow (aka port back to dazar then come back to nazj), he/she will not just show up like the bodyguards did last xpac.
you will not get credit for the three dailies. 9_9

In order to level up your follower: you will need to complete specific daily quests with your follower summoned; specifically, the daily quests which will count for this will have the text "Grants ally combat experience" when you hover your cursor over the blue exclamation point on the map. I think that these quests only become visible after you have summoned your follower.