Gold per 10 items stack with each other, however, the same gold per 10 item does not stack with itself. So, you can have a philosopher's stone and a heart of gold, and you will now have 10 gold per 10 seconds. You will only get 5 gold per 10 if you have two philosopher's stones.
Two gold per 10 items will not burn your income, especially when you got both of the gold per 10 items early on, so their effects last for a longer duration until you upgrade one of them such as the philosopher stone -> shurelia's reverie.
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!
Best Answer
The point of gold per 10 items is to increase that gold generation early since you will not be getting creep score. You get a philosopher for mana regeneration and heart of gold for health bonus for your survivability. Generally you want to build a gold per ten on a character if it is beneficial for you in what it builds into generally. Stone builds into Shurelya's Reverie, heart finally into something useful, Locket of the Iron Solari. These are both useful to a team as a support. Kage's pick only builds into deathfire, not as benificial to team, and Avarice Blade also gives gold per ten but builds into ghostblade which only gives benifits to holder again. You would want to sell pick and avarice blade later on for other items due to this fact which is not very viable. From the league wikia, the items pay for themselves after roughly 12 minutes. The gold you have to put in is locked up for 12 minutes each so there are more viable options to buy.