You need to upgrade your survey ship to a combat ship, then you'll see that it's got experience already.  Anomalies don't get you an awfully lot of that though, due to the random nature of the rewards gained for surveying.

You can "upgrade" your flagship to a warship. In order to do this, you need to research medium hulls. If you then design a warship with a medium hull, you can upgrade your warship by selecting it and then selecting "upgrade" and the ship you want it to turn into (that is also based on a medium hull). However, it will lose every feature it has (survey module, decent engines, sensors, range) unless these are present on your new design.

Personally, I have never upgraded my flagship to a warship.

dont forget to research sensors all the way.