If you've already achieved max rank, this will pop retroactively.
Syndicates are unlocked at Mastery Rank 3. You choose one as a faction and turn a small fraction of your mastery into syndicate reputation. As you max out your reputation bar, you will need to sacrifice credits and items in order to progress to the next rank.
You can earn a maximum of (Mastery Rank + 1) * 1,000 reputation per day, so for instance, a MR3 player can earn 4,000 reputation a day, but a MR9 can earn 10,000. After reaching the first positive rank, you can also do faction missions for bonus reputation that does not factor into the above calculation. With loot radar mods, you can also find syndicate medallions, which can be redeemed for even more reputation whenever you'd like.
It's best to choose a single faction and stick with it, as earning reputation with one faction will earn half as much for another, but decrease reputation similarly across two other factions. This will eventually lead in Syndicate Death Squads which show up randomly, but are much easier to defeat than the Stalker, Zanuka, or the Grustrag Three.Here is a list of syndicates.
Cephalon Simaris does not have ranks, and I am not sure if Conclave counts, so I will be writing this as if you are only playing PvE. There is no "correct" syndicate to choose. At best, take a look at their Warframe augments and choose a syndicate for one you like. Unique secondaries and primaries are also available, but be aware you need to be Mastery Rank 8 and 12, respectively, to use them.
You will need 940,000
credits in total to reach Rank 5 in any syndicate. The items that must also be sacrificed change per syndicate and are listed on each syndicate's page on the wiki. Generalized, these are 500-1,000 common resources, 2 rare resources, a Forma, an Orokin Reactor or Catalyst, a Prime weapon part, and a Prime warframe helmet. Reactors, Catalysts, Forma and Prime parts must be built, not just the blueprint.
Achievement pops as soon as you sacrifice 500,000 credits and the Prime warframe helmet.