Morgana is an excellent ap carry who is personally my favourite. They can shield themselves and snare opponents for easy escaping and they have a good damage output.

Pro's:
-Not as squishy as other ap champions
-Snaring someone then putting your tormented soil under them can do a massive amount of damage, and all you have to do is land your snare
-Has a shield for defending themselves
-Has passive spell vamp for sustainability
-Your snare is one of the longest cc's in the game, going up to 3 seconds
-If you're only a bit fed, by comboing your snare, tormented soil and ulti you can kill someone from full health

Con's:
-Sometimes targeted first in teamfights
-Your snare moves very slowly
-You can't use your moves too often or you run out of mana VERY quickly, use them more often after your catalyst

Get 1 dorans ring to start off with (nothing special), get another if you're struggling a bit. Grab boots, your catalyst, get sorc shoes then finish off your roa. If you think you're having early team fights, get your Zhonyas earlier and if you feel you're dying too much, get a rylais for extra health. When you've finished your build, just spam buy elixers.

Max your pool first, get your ulti when you can. I prefer maxing snare before your shield to burst more, but it depends if you're laning against an aggressive ap caster, getting ganked by ap casters, etc.

I personally love teleport, it potentially saves a lot of turret damage or can let you damage their turret a fair bit. Use flash (same as with just about any person).

If you're in a duo lane or don't want teleport, I'd suggest ignite as a second option.

After you level up tormented soil a bit, it will be able to kill caster minions from full hp. I tend to tormented soil the back minions so it is touching both the melee and caster minions (it just barely hits them both) and try and last hit the melee ones.

Unless you have blue or are about to b, don't use snare for farming. And even when you do have blue, don't spam it.