Thank you for the kind offer of assistance. I hope my ineptitude isn't too irritating.

Well, i've been playing against just one opponent. And usually just letting the game auto-fight my battles for me since i only have a vague idea of how the Advent ships are used. So, i've been going for a variety of ships, although i end up dead before i can get a fleet of Caps up & running.
It just seems that the AI simply does everything better than i do!

At first, i thought it was than i didn't acquire enough planets early so, so then i tried again and went out colonizing every planet i could get my hands on, but when the enemy's fleet shows up it's 2 or three times as large as mine and then it's a quick downward spiral to destruction.
Also, when his big fleet attack strikes, i have ships all over the place on different planets and by the time they reach the planet under attack, the AI seems to destroy my fleets as they arrive from different planets (and can't defend effectively as a single group).
Surely, i'm not supposed to group all my ships into one fleet?
Take a more strategic approach to your expansion. Try to keep most of your empire behind a couple of nodes. That doesn't mean you can just completely avoid defending planets in the back, but it will keep your fleet from being spread out across the entire system.
If you're having trouble defending, build hangers and broadcast centers. The first will give you pretty good defense and the computer doesn't really compensate for fighters much, the second with give you some combat buffs in your own territory.
Your fleet does not need to be "equal opportunity". Certain ships, particularly cruisers are merely support ships. They're not very good for combat and you only need a few. Try focusing more on capital ships and light frigates until you get a hang of the intricacies of the combat.
Be sure you're upgrading your planets infrastructure as you expand. Planets with low infrastructure are a drain on your resources which may be why your fleet is so much smaller.
Hope those help.