Security advice needed using hotspot

I have searched and not found exactly what I'm looking for. I am on vacation for 3 weeks away from home and using a laptop with wireless and connecting to a WLAN in a rental home at the beach. It has no encryption. On my laptop I'm running ZA (free version) and AVG.

I need to pay some bills while here and would like to use my on-line banking & bill-pay service. That site is of course SSL protected. Am I safe using the wireless to access the bill-pay site and transact business? Would it be any safer if I can connect hardwire between the laptop and the wireless router/hub?

I think I'm correct in saying that from what I've read that my data exchange to log into and subsequent business data exchange with the SSL site would be secure even over the open WLAN. However, I'm not certain and don't want to ruin this vacation with a careless mistake.

Any advice appreciated.