PHost will send you a report each turn, containing the experience levels of your ships and planets. You'll also be told the experience levels of the ships of your ship-level allies, and of your planet-level allies' planets.
You will not know the experience level of an enemy ship until you encounter it in combat.
Experience reports look like this:
(-h000)<<< Ship Experience Report >>> Id Ship Name Level --- -------------------- ----- 1 Endeavour 1 12 Voyager 2 (+) 139 Immatrikulationsamt 1 (!) 490 GORBIE CLASS CARRIER -
The Level column contains the level of the ship; it is - (dash) if the ship is inexperienced. For example, #490 is a new ship which started out with 0 experience points (but has already received points for aging by the end of the turn).
The (+) sign tells you that the ship went up one level this turn.
(!) means the ship got re-crewed and may thus have changed its level. The ship might have gone up or down, or not changed its level at all. All this tells you is that the experience points have changed beyond regular experience gain.
With ExactExperienceReports enabled, the reports will include the exact experience values.
(-h000)<<< Ship Experience Report >>> Id Ship Name Level --- -------------------- ----- 1 Endeavour 1 2274 points 12 Voyager 2 (+) 3520 points 139 Immatrikulationsamt 1 (!) 1701 points 490 GORBIE CLASS CARRIER - 25 points