The first time I went swimming in deep water, I sank to the bottom like a rock. Now that I have learned to stay afloat, I feel better about the water, but I still can't swim well.
What does "Now that" mean?
Can we use "As soon as or When" instead?
No. "As soon as I have learned to stay afloat" would be followed by "I will feel better" because it suggests that you have not yet learned to stay afloat. The same applies to "When I have learned..."
The writer has clearly already learnt to stay afloat.
Now, because I can stay afloat, I feel better about the water.
Here are some examples that might help:
Now that you're a qualified teacher, you'll be able to get a better job.
Now that I have sold my house, I can start thinking about buying another one.
Now that my kids have left home, it's very quiet!
I have a lot of spare time now that I don't have a job.