I’ve been a Star Trek fan since 1966 (yes-old), and I like all of it, including the Abramsverse. More Trek is better than less Trek.
There are three main requirements:
- A sufficiently advanced level of scientific technology. The Federation’s baseline is that the prospective member world has achieved faster-than-light space travel capability.
- One planet, one government. In other words, a system where individual nations have been unified under a single governing body.
- No form of caste discrimination is allowed.
There are two ways a planet could be considered for membership.
- The planet’s already aware of The Federation and petitions the UFP council for membership.
- A Starfleet ship examines a planet, makes first contact, and then asks the planet to join. They still have to go through the petition process, however.
Under the petition process, The Federation thoroughly examines prospective members’ culture, the rule of law, records from its central computers, etc. The whole process is estimated to take around two earth years to complete.
Please note that to fulfill their end of The Prime Directive, no planet is ever compelled to join and can leave the UFP anytime, for any reason.
In real life, some of the stories are a little murky, especially in the TOS timeline. The episode The Cloud Minders springs to mind. The Enterprise is trying to arrest a botanical plague on Merak II, and it needs zenite. The only planet with enough zenite in sufficient quantities in Federation space is Ardana. Ardana is in The Federation, and they’ve developed the capability to levitate their main city centers into their lower atmosphere.
Stratos, the capital city of Ardana
Zenite must be mined by hand, so the miners, the troglytes, are members of a strict caste system. The mines have shitty conditions. The troglytes must have transport passes to enter Ardanan cities; some have been trained to work as laborers and service personnel in the cities. It’s found later that gas emitted from zenite makes people exposed to it more aggressive.
The enterprise develops a filter for zenite gas.
This would seem to contradict what I just wrote. Well, this is TOS, where things were being worked out as they went. The only in-universe answer I could give is that Ardana faked their records and never let Federation inspectors into the mines. This probably instituted the no caste system rule and required prospective members to be completely transparent during the petition examination.