The fly spell says that you have 60 feet of fly movement.
Is there a "default height" for flying?
Can you fly 5 feet off the ground? 10? 20?
In your first turn can you fly 60 feet into the sky?
Can you choose to fly back to the ground while concentrating on the spell?
The spell Fly grants flying movement:
You touch a willing creature. The target gains a flying speed of 60 feet for the Duration.
Flying movement is described in the Movement and Position section of the basic rules:
Flying creatures enjoy many benefits of mobility, but they must also deal with the danger of falling. If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as by the fly spell.
When you have flying movement, you are free to move through the air at any altitude you like.
The only limitation is how far you move. You can move through the air up to your flying movement (which the Fly spell grants 60) at any altitude (or change altitude) so long as you do not exceed 60 feet of movement during a turn.
You may, of course, take the Dash action while flying to move 120 feet in this way. Some classes/races allow you to also Dash as a bonus action, which would increase this to 180 feet. And some effects will reduce your flying movement (e.g. Warding Wind).
If you have not moved and are standing on the ground, you may fly up to any height that is less than or equal to your flying movement (ignoring the Dash action).
You continue to fly and may fly back to the ground as long as the spell is affecting you.
There is no limit on the height you fly at. It could be 1 foot off the ground or 300 feet. The only real limit is the duration of the spell, by which you better be back on the ground or have some other spell prepared to catch you as you fall.
As for the first round, the rules for flight used to specify that you climb at half the movement rate and descend at twice the movement rate. I can't recall off hand if that still applies in 5e, but if that's the case, in the first round, if you flew straight up, you would end the round at 30' above the ground.
While flying, you can move in any direction. Unlike the flight of a bird, the fly spell itself will keep you aloft and you can change directions at will, including returning to the ground. It's not like a bird's flight where you have to be moving forward or flapping wings to maintain your flight.
There is no default height. When fly is cast, the target gains a flying speed. Using that flying speed they can use their movement as usual. The subsection on flying movement specifically addresses some issues with flying.
A character casts fly on themselves.
They now have a fly speed of 60.
On that turn, they fly up 5 feet and horizontally 55.
The character is now 5 feet off the ground.
On their next turn, they use all 60 feet of movement to go directly up.
They are now 65 feet above the ground.
The Fly spell grants you a Fly speed of 60 feet per round, and lasts 10 minutes. There are 10 rounds per minute, so that's a duration of 100 rounds. It's possible to travel 180 feet in a round if you take the Dash action as both an Action and a Bonus Action (if you have a class feature that allows you to do so), so by multiplying them together you get the maximum height attainable with a single casting of the Fly spell, if you do nothing but Dash straight up twice a round: 18,000 feet.
