I like to get a minimum of Supertonic, the Harp and any other song. Then I can use the Harp to activate Supertonic to let me use my base power to activate the other song, then finish off the Harp and heal myself. Obviously if you have the Lyra as well then you can activate Supertonic twice, and if you have a good couple of other songs out which can subsequently be activated those are good. But basically my usual style of play is to put out Arcane Cadence if I have it, ASAP, in the early game. Later on, when I have some stuff out already, I'll favour Silver Shadow over it since its effect will probably be more useful at that point. If I need more cards, Instrumental Conjuration is useful for the extra draw even if I already have all instruments in play or in-hand. Starting with neither a song or Arcane Cadence in hand can be mildly annoying because then you won't be able to use your Power phase, but you could always just draw twice and chances are you'll get something useful turn up.
Had a game earlier against Voss where none of the three characters (Tempest, Adept, and Bunker) drew particularly useful cards at the start. I did have a song in-hand but it was Cedistic Dissonance, probably the only song that's utterly useless if you start off with it, since there's no way you can both play it and use its effect (destroy an instrument to destroy any non-character card in the game) on the first turn. I did later nuke my Harp to take out the Flagship but the game was over in about four or five rounds - too much Minion damage, not enough good cards coming out :(.
I do recall a game we had, ages ago, against Citizen Dawn in which I never had more than about four cards out at once - something like Alacritous Subdominant, another song or two, and an instrument, but I was able to chain them in such a way that I could do something useful every round and we ended up winning the fight :).
Knowing other character's decks can help when playing the Adept, actually, as you'll know, for example, that Unity does well with extra plays (to get out her bots), Expatriette likes extra draws (for her guns and ammo), and so on. You can decide which character you let take which action from your various songs and hopefully get the best effect from them. I've also had a game where Fanatic had used Wrathful Retribution on her turn. On mine, I was able to get her to put it back on top of her deck (Vernal Sonata), then draw it (Counterpoint Bulwark). I can't remember whether I then let her play it again as well (Alacritous Subdominant or Inventive Preparation) or whether that occurred on her turn. Much nuking occurred, either way ;).
Finally, the one thing I give the lowest priority when playing the Adept is dealing damage. He has exactly two songs (well, four if you count both copies of those two songs) which deal damage - Syncopated Onslaught (Accompany text deals one Sonic damage to a target), and Scherzo of Frost and Flame (deal one target one fire damage and then deal one target one cold damage…or the other way round, I can't remember). I can remember one long game against…I think it was Akash'Bhuta but may possibly have been Gloomweaver, in which I dealt one hit of damage the entire game, and that was the killing blow on the villain just because I could >:). In general, I'll look at whatever thing needs to be done in the game the most right now, and then which character could do it and how, and then which effect(s) I could use to make that thing happen ASAP.
It took me a few games to get the hang of the Adept, but once I'd sort of learnt his deck a bit, I…well, got the hang of him, and he went from being "Hrmm, that guy's kind of complicated…" to being my favourite character :). Note that I don't play him every game or anything - I tend to play him in games that tend to be a bit longer, so I hopefully have more time to get out my stuff. I'm unlikely to pick him against Iron Legacy, for example, or someone like La Capitan who has a million horribel cards and breaks all my stuff :P.
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. I'm sure there's situations where an active Legacy is the best Legacy, however.