There are two obvious candidates to form Persia: The historical one Ardabil and the fastest Ajam. Ardabil has its own achievement and forming Persia with Ajam is pretty straight forward, but don’t choose one of the nations.
We choose the beast in the middle east and the king of the silk road: Timurids
But Persia is non formable by the Timis and the formable tags from the Timis are endgame tags (Yuan&Mughals). So this obviously disqualifies the Timis from forming Persia, but they have an ace: The Iqta Government to overcome this hurdle. You see why later in the guide.

You start the game like every Timurid game: Keep your subjects loyal or make them poor so they demilitarize like Germans after WW1. There are plenty of guides on the internet for this issue. Declare on Ajam ASAP and led your subjects do the heavy lifting. Classic Timurid game so far. During the war time send a missionary without maintenance to a province of Shia religion to provoke rebels. Also led Fars fabricate on Mushasha. Led the ♥♥♥♥♥♥ Rebels convert provinces for you. Take all Persian provinces from Ajam and more for 100 WS. Only state the provinces with Persian culture, integrate Transoxiana, Fars and Khorasan. State only the Persian ones from Fars. Now you have 47% Persian culture in your states, develop renaissance in them to get over 50%. Now culture shift to Persian and state the rest of your provinces. You are now in a much better culture union. Also you could swap to Shia if the rebels did convert enough for you. Take the Ifari (10% Shock Damage School). From your religion and once you unlock the national idea you have 20% Shock Damage, that’s only 5% less in comparison to a horde. The thing is you have it on every terrain.
The optimum is that you allied the Ottomans at this point.

You should be the second most powerful nation by now. The next goal is now to form Persia and finish the Timurids mission for the really nice rewards. Finishing the missions and conquering the provinces for Persia is not that hard, maybe it`s take some time if the AE is high.
Forming Persia on the other hand is much more time consuming, because you have to overcome your Timurid tag without forming the Mughals. Just remember your type of government: Iqta. The Iqta government is awesome, you can press nice buttons and you have nice bonuses, but most importantly it lets you form the Mamluks. For forming the Mamluks and overcome the endgame tag problem, you have to own Cairo and the Mamluks are not allowed to exist. There are two routes to the Mamluks: Through QQ into Syria, where you can release Syria as a really nice vassal. The other way starts in Mushasha, through the Arabian desert and ends in the Sinai. The fastest way for killing the Mamluks is allying Ottomans and Tunis. When the Ottomans start killing Mamluks on their own, it´s no problem if they have rivalled you. To speed things up you could also ally the biggest Nation in Yemen or/and the Big Boy at the southern borders of Egypt.
Like mention earlier finish your missions and kill the Mamluks. If you have AE to spare start invading India, Russia or whatever you like.

Ideas:
As your first idea pick I advise you to pick religious ideas if you converted to Shia, because you have to convert so much stuff. It costs some money, but you will always have 100 Prestige so you can disinherit every time when your heir is trash. Just throw him from the cliff like the Spartans.
The later choices are up to you: Diplomatic, Quantity and Admin are always good. Offensive has a really good polices with religious for converting these pesky Sunni heretics. Aristocratic ideas have also a good policy with religious and is always a really good allround choice. If you want max out your advisors early in the game and steal the money from the Indians, then take trade ideas.

Now you should be able to form the Mamluks. By forming the Mamluks your capital moves to Cairo, so you are now allowed to form trade companies in Asia, which is op considering your easy access to India. After forming the Mamluks spend 200 diplopoints for moving your trade capital back to Persia. Now you could finish the Mamluk missions for even more rewards or form Persia right away. If you want to spawn global trade in Persia with ease, conquer the Aleppo node, so nobody can steal your stolen money from the Indians out of Persia.
By forming Persia instead of Mughals as Timurids, you may not have culture assimilation. Trade companies make more than up for that. Who doesn’t like trade companies?

If you compare the available idea sets for you, I would stick by the Timurid ideas: They have the strongest military with 10% Morale, +1 Land leader shock, 10% shock damage and 5% discipline and they have corruption reductions which is really nice when you blob. Mamluks have the weakest Military, but the strongest economy with bonuses to trade. Persian ideas are an allrounder: Strong enough military to stomp the Ottomans with a decent economy (Goods produced).

By forming Persia and finishing Timurid, Mamluk and Persian missions while enslaving the Indians you will be Great Power #1 by far. With the Persia node and trade companies your economy will be turbo charged, so you can max out your monarch point generation. Get the achievement “THIS IS PERISA” and then do whatever you want. Your Persia will truly be a Super Saiyan before 1600.