Generally the most simple information you'll get about enemy ships is from the exact first contact you described, Any passive or active sensor that detects the ship would tell you their speed, Thermal sensors would detect their drive size (you can estimate tonnage from drive size and speed you need to guess their tech level though), EM sensors would detect their active sensor or and shield strength, active sensors would detect the tonnage, when your ship explodes you can usually determine how it's armed, what strength warhead etc if you include a small resolution 1 sensor you could also find out the speed of their missiles (as long as it can detect the missile at least a whole increment before the ship explodes.
More information requires prolonged contact with the enemy, you can usually find out the enemys maximum speed if/when they run after you try to intercept it with high speed fighters or FACs, or even your other ships if they happen to be faster than the enemy. Note they'll usually only run if they're out of missiles and arent beam armed.
I believe you can get other information that shows up on the intelligence window but I'm not too familiar with it.
TLDR: you need bigger (more costly) sensors