Some 1 round spells, e.g. Blade Ward, have "Until the end of your next turn". others, such as Ray of Frost, have "until the tart of your next turn". Some, such as Chill Touch, even have both!
That is, a 1 round spell, cast during your turn on round 1, generally lasts until your turn on round 2 (1+1=2), but whether the start or end varies.
However, for longer durations, such as 1 minute (10 rounds), the section on Duration (p. 203) does not specify a general rule.
A good rule of thumb for GMs to follow, for a spell that lasts 1 minute, is to get 10 turns worth of benefit from it.
So for Haste (following this rule of thumb), which gives you an extra action and extra movement, including on the turn you cast it, as well as an AC boost, A GM may allow it to last until the start of your 11th turn.
You get to take an extra action on turn 1, and turn 10, but not turn 11; the AC boost protects you after your action on turn 1, until after your action on turn 10, right up until you start turn 11 (but not during your turn 11).
In contrast, a GM might allow Bless to last until the end of your 11th turn, so you get 10 turns of benefit from it (rather than only 9)... Or they might decide, for simplicity, that long duration spells are always the start of turn 11.
In both cases, a spell with a duration of 10 rounds, cast on your turn in round 1, should last to your turn on round 11 (1+10=11), with the GM determining if it is the start or end of your turn 11.
Suggestions applying the rule of thumb to the two spells you mention:
For Aura of Vitality, as you could use it on the turn you cast it, I would allow it to last up until the start of your turn on round 11, although in practice it has no effect after your 10th bonus action during your turn on round 10.
For Moonbeam, it mentions using an action on each of your turns to move it; so I would allow 10 actions, until the end of your turn on round 11. The would allowing you to move it on your 11th turn, but doing so would be redundant as it would cease at the end of your 11th turn (just after you move it). It would potentially affect enemies 10 times: their first turn after you cast it (i.e. the rest of round 1, up until your turn in round 2), right up until your turn in round 11.