Now I think it might be
STEAM.
I go in a cup,
but you can't drink me up,
A cup of tea or coffee might be steaming, but you can't drink steam.
I soar through the air,
well above your hair,
Steam can pass by well above the level of a human head.
At first I have need
of some serious speed.
Later, it's nice
to be more concise.
Steam might start off flowing quickly but end up dissipating in the air.
To the club we could go
if it doesn't snow
If I go to the beach
You will probably screech.
I haven't worked these lines out yet!
First attempt
Could it be
a STORM?
The title suggests we might be looking for something in the natural world, and
a storm is certainly a force of nature.
I go in a cup,
but you can't drink me up,
Refers to the well-known phrase "a storm in a teacup".
I soar through the air,
well above your hair,
Stormclouds pass by well above the level of a human head.
At first I have need
of some serious speed.
Later, it's nice
to be more concise.
At first I thought this was wordplay like in the OP's last riddle: I was looking for words like fastshort. Now I think it might be more literal: storms start off fast and violent but later calm down.
To the club we could go
???
if it doesn't snow
snowstorm.
If I go to the beach
You will probably screech.
sandstorm.