Hi Cleiton,
Based on - this is usually used describing assumptions, or proposed facts. It means that from what can be seen, something can be guessed.
For example, "based on your question, you're studying some more advanced English structures". I don't know this for sure, but from the evidence in your question I can assume it.
Rest on - this is more often used for more secure support of a fact (rather than to qualify something).
For example, "our projects rests on copious research." We know the project is good because it rests on a solid basis of evidence.
Rely on - this is used more to show a support for behaviour rather than thought, and thus is often used socially.
For example, "my brother relies on me to drive him everywhere." This is more of a dependence that could be unpredictable (at any time I can stop driving him).
Lean on - similar to rely on, but describes something strong that supports something weaker (the weaker object does the leaning).
This can be either physically - "the boy was leaning on the wall"
or emotionally - "lean on me when you're not strong, I'll be your friend"
They are really similar in meaning, the real difference is in the use.