Proof by contradiction confusion I am having confusions on some basic stuff. Could you help me out, please?

Consider the following statement:

"If A then B or C"

If we apply proof by contradiction method to this statement then we assume the contrary of "B or C" which is "not B and not C". Than if we deduce "B or C" by using "A" then does this count as a proof?

Thank you!
Let's say you have to prove
$$A\Longrightarrow B $$
The thing you are looking for is: Assume that $A$ is valid and assume for a contradiction that $B$ is invalid. You would then consider
$$A\land \neg B $$
If that yields a contradiction, then the assumption that $B$ is invalid, is incorrect. Hence $A\Longrightarrow B$  

Another way is to prove by showing the contrapositive is valid i.e $\neg B\Longrightarrow \neg A$.  

In your case we consider:
$$A\Longrightarrow B\lor C $$
Assume for a contradiction that $\neg B\land \neg C$, then it must be true that:
$$A\land\neg B\land\neg C $$

You had suggested a proof by:
$$A\land\neg B\land\neg C\Longrightarrow B\lor C $$
You will no make progress, for we can apply your method indefinitely and arrive at:
$$A\land\neg B\land\neg C\land A\land\neg B\land\neg C\Longrightarrow B\lor C $$
Which is the exact same thing we started with. Stuck in a circle.

The problem is that you are using the validity of the very thing you try to verify as part of your assumption.
The proof by contradiction in this case would be to show an example where "A" is true but neither "B" nor "C" are true.