There's the main SMT series, and then the spin-offs. Only two games are direct sequels: Digital Devil Saga: Avatar Tuner 1 and 2 and Persona 2: Innocent Sin and Eternal Punishment. Most games have some kind of reference to other games, but you don't have to play all the games to understand the others.
Unlike in most RPGs, stuff like weaknesses and buffs and debuffs matter. It's easy to win a tough fight with low levels if you play your cards right, and it's equally easy to lose a tough fight with high levels if you have no strategy. You can negotiate with demons and get them to join you, and then fuse them to create new demons, so it's different from the normal RPG series.
I love the SMT series, it ditches a lot of the typical RPG fantasy stuff and has wonderful writing and music. The strategies tend to be more in-depth and there's a huge degree of customization to be had.
MT and SMT games - typical dungeon-crawler. Fuse demons and go through dungeons. SMT3 is often cited as the game you should start with, though it's quite a tough game. Strange Journey has a sci-fi element, if that's more your thing.
Persona - Persona 1 and 2 retain core SMT gameplay, but instead of demons you have Personas (they're basically the same). Persona 3 and 4 still has fusion and dungeons and bring in a dating sim-esque element. I recommend Persona 2: Innocent Sin the most.
Devil Summoner - First two are core SMT, but add a "Loyalty" element. The 3rd and 4th games - the Raidou games - are action-RPG and feature real-time battles. Try out both types!
Digital Devil Saga: Avatar Tuner - completely throws demon negotiation and fusion out the window, because your characters are demons. You learn new abilities and skills via a "Karma System," so pretty much any character can learn any skill. Highly rec'd!
Devil Survivor - more Fire Emblem than SMT. There are no dungeons, only pre-set battles. Get new demons via auction and train up in "free battles." I've only played the first one, but it's great.
There's also the Last Bible series and Demi-Kids series among other individual titles, but those aren't really like the other games and many aren't translated.
Bottom line: Very good games. Start with SMT3, SMT4, Digital Devil Saga: Avatar Tuner, Devil Summoner: Soul Hackers, or Persona 2.