The Adept is my favourite hero :). Here's how I tend to play him…
Early on in a game I will always play Arcane Cadence if I have one (and chain it into another if I get one…today in a game I managed to chain all four :D), as this is the best way to get through your deck.
It's good to get a song out on your first turn, just so you can actually do something in your Power phase.
My "uber combo" is the Harp, the Lyra, and Inspiring Supertonic - one instrument chains into Supertonic, which chains into the second instrument, which chains into Supertonic again, which chains into a third instrument (or my innate power if I don't have one out), which activates whatever it can if I have anything out that's useful at the time, then it goes back along the chain to finish off the second halves of all the instruments and nets myself at least 2hp in the process since I always use the Harp's second part to heal myself with Supertonic.
Just in case you're not aware, Performs and Accompanies aren't powers, so you can activate the same song multiple times in a turn if you have the means to do so (see the Supertonic combo above).
I tend to use Silver Shadow later in the game, when I've got some stuff out and don't need to use Arcane Cadence so urgently.
If you have to destroy an instrument (either via Cedistic Dissonant or via a villain or environment effect breaking it), it's not actually that much if a problem since instrumental Conjuration can search your deck and your trash, so you can get your gear out no matter where it's ended up. Annoying if you find yourself combo-less for a while but at least you have the means to recover.
The Adept is good with anyone because of the way he can let people do more stuff, but some heroes in particular work really well with him - Unity can't play bots in her Play phase, so really appreciates the Adept letting her do so when it's his turn. I also had a really good game with Captain Cosmic once where he put one of his construct thingys on me which let me use a power every time it took damage (and then other heroes kept deliberately plinking it for single points, only for me to heal it back up again via Rhapsody of Vigour at the end of the resultant power chain :D). Anything which lets the Adept use a power out of turn is good, especially once he's set up.
It does tend to take the Adept at least a few turns to get going, so he tends to be a bit less useful in shorter games, but even with a couple of songs and an instrument he can he super-useful, just by buffing someone's damage or letting them play a card or something :).