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My mom asks me whether I want to go to my friend's house and I said yes.
Meine Mutter fragt, ob ich zu meiner Freundin gehen will und ich antworte mit "ja".
What is the formal way to say "to say" in German? In some books I found that the verb 'erwidern' is used more often than 'beantworten'. Is erwidern more formal than beantworten?
Erwidern is quite formal (it is hardly ever used in spoken language outside of legal or political discussions, more commonly encountered in written form). It means to reply to a question or a statement. An alternative that shares the same basic meaning would be entgegnen, which I would also consider rather formal. Antworten is certainly the most common way to express that someone is responding to a question.
In addition, erwidern is also used to more generally express that someone is retaliating or reciprocates someones actions, e.g:
Die Soldaten hatten den Befehl, im Falle eines Angriffes das Feuer zu erwidern.
They were to retaliate in case of an attack, but not allowed to fire first.
Sie erwiderte seinen Blick.
They exchanged a glance (some kind of mutual understanding is implied here).
Er erwiderte ihre Gefühle.
He returned her affection/had the same feelings for her.
The formal way of "to say" is... "sagen".
This word is entirely fit for any context. "Erwidern", "entgegnen", "antworten"... all those are speech tags too but they do not mean "to say" but rather something along the lines of "to respond".
In novels authors often use direct speech and they have to use the verb sagen very often. So it is natural that you will find a lot of variants for "sagen" used to avoid repetition. A variant such as "erwidern" has nothing to do with formal or not formal, it is simply a literary variant for sagen.
... und ich bejahte.