So, I just took a look at the hands for the last game we finished, since it's the only one I can see.
In terms of cards, I see a few things:
You've still got a lot of the starting cards (apprentice, militia, mystic, heavy infantry). I banish those ASAP (apprentice/militia first).
You have a lot of lower value cheap cards like Prodigy, etc. Generally, I never pick these up.
You don't have many card drawing cards. These are critical, and the core of your deck building.
You don't have many centre row banishing cards (most of which are Enlightened). These give you a higher chance of getting decent cards on your turn, and almost always draw cards as well, feeding the previous point.
There's too much focus on monster killing. It's too random as a strategy, because you can't tell if the good monsters will be available to kill on the turns you have a high power score.
In terms of value cards, I look at them like this:
Green Point/Mana + Draw cards - Lionheart, Witches, Starchild
(Early game if at all possible) - Card banishing cards. Rai, Journeyman Apprentice, Black Hole, Void Apprentice, Deathsworn Warrior, etc. Getting rid of terrible cards lets your good ones appear more often.
Card drawing cards - Almost anything Enlightened, anything else with card drawing.
Mechana Constructs - often not that useful, but worth a lot of points
Reasonable filler cards when you can't get any of the above - Grangers, Heroes who don't meet the above requirements, Void Avenger, Brazer Drone, etc. This one is a little catch as catch can
In terms of strategy:
Delete any crap cards ASAP
Grab a few mystics (usually max of 4, occasionally more)
Buy cards listed above
The only time I venture off-track is those games when there's a ton of red at the beginning. Usually have to grab a bunch of infantry, and then I do worse because getting to the card cycling is disrupted.
And I don't actually play the game that much, or normally under the same rules - I usually play 120 point games solo, and will often lose to the hard AI under 60 point rules because it brute forces points early using a heavy infantry spam with a few mixed in decent cards. Over the longer points I can build an engine to squash it reliably.
At some point, I should get the new decks/expansions, but I don't think the base strategy changes much - it's just how the rules of Ascension are written, or indeed any card cycling game.