Been playing a little bit of 
support and it seems like she is quite strong. I'm a gold 4 support main, and I build her like so..
-> 
-> 
-> 
-> 
-> defense item
Her W is incredibly reliable at procing spellthief, at a very cheap cost as well. I find that I can reliably harass with minimal mana problems.
Does she have weaknesses, and how can I get gud with her?
I think Morgana is definitely an underestimated support. Proper black shield timing can negate CC and keep your carries alive even if they are out of position (and we won't even mention how ridiculously long her binding lasts.) However, she is very immobile and very squishy which makes her incredibly dependent on Zhonya's in team fights. In a long laning phase I think you build could work but if towers go down fast and teams start to group up for objectives quickly you may need to rush Zhonya's earlier.
I'd go with MR or Mobility boots. You don't really need the damages as support, just all the snares, and the extra move lets you set up the ult better. You're at a big disadvantage if you get caught out with everything on CD. Don't forget about the 
if you want your teammates to hit like that Daft Punk song. Also, keep an eye on that ult; if someone is hiding near you (in a bush, or 
etc) it will light up and you can get em.
Randomonium already beat me to the good advice. I don't even screw with the top upgrade for FQC, I just go straight for Zhonyas if I don't need something else more. Mobility is a weakness and squish, but she's not as squishy as some. That binding is also slow as a truck and easy to dodge. Make sure you're shooting forward/back, not side2side, and that you're dropping a W on people that get caught in it, and make sure you're leading them. Also, it goes through stuff, so you can pop them around corners, or even steal a dragon with it if you're quick enough.
Welcome to the dark side!
I played sometimes Morgana support, both in normal and in rankeds, but ended up struck with Zyra concercing magelike-supports, but i believe to have grasped some experience.
One thing i have learned is that Zhonya is really really necessary in teamfights: is what makes her really temible, because when the enemy engages you can use R and Zhonya to counter-engage safely; the timing depends on the situation, for example if an assassin targets your carry you might prefer to black shield him and snare the assassin, then using R. You can also use it as an engage tool, mostly in crowded areas like dragon pit and so on, always with great care. For this reason i suggest you to pick it as soon as you complete your "support duty", like getting sightstone and so on.
I don' t advise you to pick sorcerers unless you are really fed or need a magic damage source, they cost a lot and for teamfights you need more Zhonya, plus for roaming, warding and chasing mobility are more useful for a support, since they are also really cheap.
On magelike supports i love 
Banner of Command because helps my team with a MR aura while still giving ap to the champion, plus a very good active to asset some lanes that your team mates still refuse to push.
I would not advice to pick rabadon, again, unless very fed or unless you really need a magic damage source.
I am only a silver scrub, but if any of what i do say makes any sense glad i could help.
Tips on Support Morgana gameplay:
.
who can just Q-AA(passive) to wittle down your level 1 E fast.Build paths:
: An excellent support item choice for Morgana.
: Not really needed when you can heal off & do ez Spellthief pokes with your W.
: Upgrade it to this as soon as possible, but depends on the amount of gold you gained. Try to recall back when you reached at least 1300 gold, so that you can buy both at the same time. If not, upgrade your Spellthief first to gain gold easier.
: Buy this boots in order to have your abilities come off cooldown as fast as possible, in terms of catching off enemies & saving your allies once in a while.
: Only if you're going on the offensive, or your team lacks the amount of AP damage.
: Just fucking buy it. No need to explain this. Upgrading is totally optional.
: This is a must have item after at least Sightstone/Frostfang. You'll definitely need it to pull off a more successful ult.
Morgana can have two build path, namely the AP builds or the Support Tank builds:
AP Builds
: Builds more AP. No shit.
: Grab it when enemies have more AP damagers & to sustain your mana well since you have to spam your abilities lots, especially if you're building .
: If you don't have much of a problem with mana sustain, needing CDR & be more offensive, buy this. Works very well if going against an annoying tanky 
.
: Only if enemy is investing heavy MR builds.
: Not so much of an ideal item for a Morgana, even though your E does procs the passive. Build it only if your team is fully composed with AA-based champions.
: You don't really need it when your passive already procs the Spell Vamp.
: An ideal item if you wanted some slows & offensive stats.
Support Tank Builds:
: A perfect support item when enemy team is composed with AoE & AP damagers.
: A must have whenever your team wanted some engage.
: A good item to protect your squishy carry under heavy CC. Also can be a good item to proc an emergency heal for your team.
I always alter my build path depending on my income. If youre doing well (winning lane) and can afford it, then going for a mage build is a great way to help your team keep whatever lead you may have/get.
However, if you don't have enough gold coming in it becomes difficult to buy all those super expensive AP items in time to be relevant, so your best bet in this situation is probably going to be building more to support your team with things like 
and .
I'm only a bronze 1 (1st season) barely played any ranked, but have played a fair amount of morg just got lvl 5 with her. She has a really flexible build path in my opinion. Obviously Zhonya's is conisdered a must in most games, but is expensive. Contrary to what most people say I actually prefer coin builds in some matchups with her for the added sustain in lane and cheap mobility boost for immobile adcs to help escape or close in for picks and binds from fog of war. Gives her a bit more safety in her roams and warding along with mobies which are more cost effective. This all depends on your play style though. If you snowball CDR such as Morellos is a good option. It adds a poke factor to her game which can help all lanes as opposed to just bot. Honestly just like spellthiefs for checking brushes, but understand why people play it more frequently than coin.
Decrit gave you good advice; Banner of command is a great item on ap supports and Rabadons is not a good choice unless very fed, it's a very selfish item on a support. Your main role as a support is utility, not damage and that 3.3K gold would be better spent on an item that gives you that like , , , 
, and even 
.
Focusing on CDR and MS is a great idea to utilize your bind, black shield, and ult stun.
As for weaknesses, and this is just from playing against them in bot lane:
As a supp her most notable skill is her Q. It's an amazing source of engage for your team and great at punishing players who are alone or out of position. Morgana support is a lot about timing with her black shield (particularly is ur vs a hook) and Q.
Also, learn the best way to use your soil in lane. Use it to zone the enemy ADC (they don't want to give you those free spellthief procs) and poke them down. Use it to push the minions faster if you're trying to back or they're under your turret, that kind of stuff.
