As Drmixable mentioned, it is very situational. If you're intending to solely be a kill lane, it might make sense to go Sapphire Crystal, ward, health pot, then tears.If you do this, you are gambling completely on your lane getting first blood and then compounding that advantage into laning dominance for the rest of the laning phase.
If you are wanting to be a more traditional support early game, then definitely do the standard Faerie Charm, wards, health pots mix.
Even if you do start with the mana crystal and go straight into tears, I think it still makes sense to move towards the more traditional support build after you pick up the tears. Philo stone, Heart of Gold, lots and lots of wards, oracles, etc. Build your gp10 items into their upgrades, prioritizing Shurelya's if your team is winning team fights and Locket if they are not.
And - as always - buy boots on your second trip back to base. For Blitzcrank, Boots of Mobility upgrade later makes a lot of sense for getting into position to yoink someone or knock them up to protect your carry. However, if the team is full of hard CC you will of course want the tenacity from merc treads instead.
Other good items include getting an early catalyst(which will build into Banshee's later), to sustain your lane harass and increase your passive shield proc if necessary. An aegis, of course, is always welcome from a support. Frozen Heart could be viable if the enemy team has more than two sources of straight auto attack damage, but generally won't be worth it.
Hopefully this wall of text gets you started on the path to thinking about how to apply your items to each situation. There is no single item guide that any hero should follow. Anything from who you are facing, how much gold you have, and how late it is in the game can vary your items and the order you build them in.
GL HF!
You can play almost any champion as a support, but there are a few things that are important for support champions to have:
Peel. This is one of the most important traits for a support champion to have (although supports with poor peel can still do well). Most types of crowd control are good for peeling, especially knockbacks and knockups. Some good examples of very strong peel supports are Alistar and Janna.
Sustain. This can be in the form of shields or heals, as long as they can target allied champions. This will allow your lane partner to stay in lane longer. An example of an excellent sustain support is Soraka.
Utility. This is extremely dependent on the specific support, and can mean a lot of things. This encompasses things like movement speed buffs, attack speed buffs/slows, and so on. There are many high utility supports, such as Sona, Nami, and Thresh.
Poke. A support with strong poke is able to instil fear within the enemies and force the enemy to trade CS (and thus gold) for damage. Good poke supports include Sona, Thresh, and Lulu.
Zoning. This is less applicable to passive supports like Soraka, but zoning is an important trait for support champions. The ability to scare off lane opponents denies them gold and experience, making it easier for your lane partner. Good zoners include Blitzcrank, Thresh, Taric, and Leona.
High base stats/damage. Champions with high base stats will have an easier time in lane, and champions with high base damage can support and still deal relevant damage without items. Examples of this include Zyra, Fiddlesticks, and Jayce.
In addition to the traits of the champion you select, you must also be mindful of your own characteristics. There are a few things you should do to be a good support:
Do not take farm from your lane partner. They usually need it more than you. The exception is when they certainly cannot get the farm (e.g. they are in base) — in that case, it's better that you have the gold than nobody.
Be careful about pushing the lane. When you push up, you render yourself more prone to ganks. Of course, as the duo lane, if you've been warding well, you can do this with slightly more safety than other lanes — but just because you can do something doesn't always mean you should.
Stick to your carry. Do not chase and ditch your partner. Exception: if your carry will die for sure, just run. If, however, you can sacrifice yourself to save your partner, strongly consider it; it's often much better to have a carry alive and a support dead than the other way around.
Ward efficiently. This doesn't necessarily mean ward everywhere; just be sure to ward the river, any gimmicky spots (for instance, playing support against a Purple team with a jungle J4 means you should ward the river brush), and the lane brushes. Do not be afraid to purchase and use Vision Wards to make sure the enemy does not have vision of gank paths, or if you're a zoning support, the brush you should be camping in.
Be very map aware. Know whether any enemies are missing, and don't hesitate to call mia/ss for other lanes. Ping your allies if you notice enemies ganking them, since they may not be looking at the minimap.
Think about your purchases. Aside from the standard Ruby Sightstone that every support should build, your items should depend heavily on the game. Don't just blindly follow a guide; put some thought into what your team needs. Strong items for almost any support include Aegis of the Legion, Locket of the Iron Solari, and Shurelya's Reverie. You should also try to always carry Vision Wards for vision control.
Best Answer
It depends on what your full build will look like. Essentially Ruby Sightstone and the three Eyes are bad full items, they give you some HP plus the extra ward, but they aren't that good. That is valid for all three Eye items (Supp Items at the latest stage, combined with a sightstone).
They however provide something extremely useful, an item slot. Should you go for the Frost Queen's Claim, Talisman of Ascension or Face of the Mountain (The non-warding counterparts of the Eyes), you would get useful item actives, with the drawbackbeing that your Sightstone would take an extra spot (And in the end you could only upgrade it to Ruby Sightstone).
Essentially whether your go for Ruby Sightstone or an Eye item depends on whether or not you're going to use the actives of Frost Queen/Talisman/FoTM. If you are going to use them, you're unable to purchase the eyes. If you're not going to use them, the eyes provide better overall stats.
Now as per which of the Support items you're going for, that depends a lot. If you're thinking about late game, the Talisman allows you to get move speed for your team, while the Frost Queen gives you 2 ghosts who locate enemy champions (and slow them); The Face of the Mountain provides a shield that scales with your Max HP to an ally.
Essentially, if you think about the late game, you would pick the Talisman if your team needs movement speed to engage/disengage. You would pick the Frostqueen if you need vision or need to catch the enemy team unprepared. If you're planning on being massively tanky, the Face of the Mountain would help you save your teammates/yourself.
However if you think about the early game, it depends on what you'll do in the lane phase. Ancient Coin heals you by being enemy minion deaths, while giving you gold for it. Spellthief's Edge is useful if you'll be poking a lot (you gain gold from hitting with spells/auto attacks). Targon's brace heals you and the closest ally when you kill a minion (and have a stack), while giving both of you the kill gold.
In Sum:
You go for Ruby Sightstone (as a last HP item) if you purchase Frost Queen/Talisman/FoTM. If you don't go for either of those 3 legendary tier items, then you should go for the Eye of the basic item you purchased, in which case you should see what brings you the most advantages early on.