No, you don't need to set up a business account.
Just keep a record of your income. You will have to make a Self-Assessment return to HMRC, on which they will calculate the tax due.
As the above suggests you can offset some expenses against your taxable income. Seek further advice from HMRC, and not my job to give you financial advice, but look closely at what you can claim in mileage rate if you visit customers to give lessons, things like servicing/tuning your piano, and claiming household expenses - which from memory is �3 pw without having to justify it in receipts. All helps.