Anyway, it's correct at the beginning of a sentence, don't worry. It's not something that "has always been said that way" - it's actually grammatically valid. Anyway works as a transitional adverb and can go at the start of a sentence without any issues, maybe followed by a comma for clarity ('Anyway, I'm not coming to the party'). Your mom probably thinks it's too informal or sounds off, but that's personal taste, not a rule.
The thing is, anyway has a slightly different meaning from however or but. However and but introduce a real contrast ('He wanted to come, but he's tired'), while anyway signals more of a conclusion or continuation regardless of what was said before ('I'd like to come, anyway I'd rather stay home'). It works well in more conversational and informal tones, which maybe your mom perceives as less "correct," but it's correct anyway. It's just that all three connectives have slightly different uses.