Joining the Fighters Guild is pretty simple. Just speak to an appropriate member at any of the guildhalls (they are in all of the major cities, except the Imperial City, but only Cheydinhal, Anvil and Chorrol have quests). Assuming that you meet the requirements, you'll be made an Associate. That means you now have a place to sleep in every town and have access to the talents of the guildhall steward for repairing your equipment. The guildhalls are also a decent place to pick up low-level equipment at the beginning of the game (at later stages you'll be able to find much better equipment as loot). Anvil is also a very good place for finding low- and mid-level trainers in the various combat skills. Don't forget to search everything at the guildhalls. You should be able to turn up one or two skill books at each of them.
Aside from not having a current bounty on your head, there are no requirements to join aside from asking. I have heard that having an Infamy score higher than 100 could be a problem, but have not verified this. I guess a quick way to find out would be to complete the Thieves Guild questline before joining. Since the Cowl of Nocturnal gives you an Infamy of 100 while you are wearing it, that would pretty much clear it up.
Fighters Guild quests come in two pieces or chunks. During the first piece you complete contract assignments given by Azzan in Anvil and by Burz gro-Khash in Cheydinhal (ask about "contracts"). Once those are done you go talk to Modryn Oreyn in Chorral (ask about "duties"). Then it's back to Azzan and gro-Khash for some more contracts and then back to Oreyn for duties again. You'll do this process four times before you're done.
Similar to the Mages Guild questline, the overarching story for the Fighters Guild is the elimination of a rival mercenary group called the "Blackwood Company". You'll hear about them in your first set of "duties" quests, but won't have anything to do with them until much later in the questline.