Do I need to download Qiskit separately or can I learn and write programs as well as practically view possible visualizations in IBM Quantum Experience itself?
You can use so-called Quantum Lab in IBM Q Experience web interface. Log into your account on IBM Q and click on this icon:
After that, click on New notebook button and here you can write your code in Qiskit and run it.
The Quantum Lab is based on Jupyter notebooks and interactive Python.
Also, you can use the Qiskit tutorial which is a part of the Quantum Lab.