funds to a new address)?
Are you talking about a normal transaction or about the change of a transaction?
The change of a transaction simply just is an additional output. So an usual transaction has at least 1 Input and 2 Output (Receiver / Change).
Each transaction requires a fee (based on the size) to be included into a block. It doesn't matter whether you are sending the coins to an address owned by you, or someone else.
Noone except you knows that you are owning the receiving and sending address. A transaction between 2 of your addresses is a normal transaction for everyone in the network.
You will be able to choose the fee, just like with each other transaction inside mycelium.