What, if any affect, will there be to my credit score by using a balance transfer(for lower interest rates), as opposed to simply paying the balance down normally?

Paying down a credit card debt leaves you with less outstanding debt than if you transfer the balance to another card. Consequently, it is better for your credit score to pay down the debt rather than transferring it.

If you transfer to an existing account, there shouldn't be any effect. If you transfer to a new account there may be an affect since the available credit goes up.

My goal is to primarily reduce total interest and secondly to improve my score. By simply paying down, I subsequently incur less outstanding debt, without decreasing the applied interest rate. If I transfer to a lower rate card and pay it down, I would think will increase my score, but it does leave a card with no balance...