I have a credit card with a high interest rate and would like to transfer my balance to a credit card with a lower interest rate so that I can pay off more principal and a little more quickly towards being debt free. I understand that applying too often for credit cards can damage my score. If I'm not approved for a credit card from one institution, how long should I wait before applying for a card from another institution?

Depend open credit card providing company..

You should apply for them quite close together. Those inquires will be on your report for one year, however it takes 30 days for them to show up for the first time.