Cancel-casting is a technique that pays off for Priests more than for any other healer because Priests have ways of keeping out of the 5-second rule while still healing (Inner Focus, Holy Concentration procs) and have several talents that reward stacking Spirit. When Holy Concentration procs, you cast your next Greater Heal for free, so you can land a huge heal and not have to worry about the tank too much for a time. After that, you can hit Inner Focus, so when you do have to heal the tank, that heal's free too. By cancel-casting the Inner Focus heal and the next, non-free heal you cast, you can stretch your time out of the five-second rule and bank a great deal of mana. If at the same time you trigger spirit-boosting trinkets and the like, the effect is nearly as dramatic as Innervate.
But if you're not raiding, chances are you won't be in any fights long enough to really see the benefits of cancel-casting. I've never been in a heroic fight where I couldn't keep healing just by using a well-timed shadowfiend and perhaps a mana potion. Your challenge in heroics will simply be learning to choose the right tool from the enormous Priest toolbox for each fight.
Prayer of Mending is a no-brainer. You will cast it on your tank every ten seconds. Renew is another fire-and-forget heal that you should keep ticking on your tank. It's also good for topping off DPS who take a little splash damage, life-tapping locks and the like. You can even use downranked Renew for your DPS to save a little mana.
You should have max rank and a downranked Greater Heal on your bar. This is your most efficient single-target direct heal thanks to casting coefficients and Holy talents, and of course max rank will be overkill at most times. If you're not casting another heal, you should be cancel-casting this on your tank.
Don't bother switching targets when you take damage. Just start casting Binding Heal. It's fast, low threat and insanely efficient. You can keep yourself alive through any AOE or splash just by spamming this, and it's awesome on fights where you can expect to pull aggro but not to get two-shotted, such as the PVP-style fight in Magister's Terrace or any boss with lots of adds, like the Underbog's Black Stalker..
Prayer of Healing is simply amazing when the whole party is taking damage. I don't know how the heck a Paladin heals through the Infernals' meteors at the end of the Arctraz, but Priests have it easy. When you see the big rock coming down, start casting PoH. Mischief managed.
Power Word: Shield is great for proactive damage prevention. Cast it before a pull on the DPS you expect to pull aggro, such as the AOE mage or warlock to help keep them up and to prevent spell interruption, or on your rogue to give him a better chance of surviving to Vanish if the Sap goes wrong, or on CC Succubi to keep them from being interrupted by splash damage, which usually leads to a short Seduce and a dead demon. It's also the oh-crap move you can pull when someone is likely to die in less than 1.5 seconds.
In all other oh-crap situations, there's Flash Heal. Rightly scorned for its inefficiency, this spell is nevertheless what you will turn to when the tank's health is critical. Don't chain-cast it, however: get back to Greater Heal as soon as you can.
Circle of Healing is amazing in high-end raiding, but pre-raid gear doesn't really support it and I haven't found a fight in Heroics that called for it. Even fights where you need a lot of mobility and there's a lot of splash damage--the Astromancer in Mechanaar for example--are easy enough with a mix of instants (PoM, Renew, PW:S) and PoH, and your peeps tend to get so scattered that CoH might not cover them all.
I've never used Lightwell or Holy Nova, so I can't reference them. They seem to be the suck.
Finally, don't forget your bandages! When you're conserving mana and stretching your time in the 5-second rule, nothing like getting a few bandage tics on your DPS or yourself.
In PVP, your role as a discipline priest is actually that of an offensive/healer hybrid. You'll master the art of doing the minimum healing necessary while aggressively dispelling the enemy and debuffs that are keeping your teammates out of the fight, looking for an opportunity to mana burn enemies, and using your fear to maximum effect. The time to cast Greater Heal is a luxury you won't have; you'll be relying on Renew, Prayer of Mending, Pain Suppression and Binding Heal, plus Power Word: Shield when appropriate (Reflective Shield makes it often appropriate!) and Flash Heal on rare occasions. I recommend a Mutilate rogue partner for 2s, a Frost Mage and a Shadowstep Rogue for 3s, and any comp with a main healer who is not you for 5s.