There are some obvious choices for both (Fighter for Leonin and Bard for Satyr), but was curious to see how others are building these races.

Satyrs are a great fit for Paladin, especially Oath of the Ancients. Leonins are too generic a race, we've already got a Fiend Folio's worth of races with bonuses to Strength and Con and Lion-O Leonins don't really bring anything to the table that's that different from what you can get from a Mountain Dwarf, Half-Orc, or Minotaur.

Yeah i hadn't though of Paladin for Satyr, but that would be awesome. Ancients goes well with the fey theme too. Valor bard or heck even Archfey Warlock would synergize well with base stats. 

I actually think Leonin are a fun flavor, probably closest comparison would be combo of Minotaur and Half-Orc. They have Natural Weapons, Darkvision and Fear future you can use a bonus action, choice in prof bonus (all useful) and extra movement speed isn't too bad. I was hoping for different Stat block (Like STR & WIS or CHA), but I think any melee class could use them but was asking what subclass could benefit from the racial traits the most. 

Natural weapons are cool and all, but unless you're in a situation where you for whatever reason don't have any normal weapons they're not really all that useful.  In all the time I've been playing 5E, I can still count the number of times that that's happened on one hand.  I suppose the racial fear-inducing ability is useful for a Paladin of Conquest.

I'm with 6th here. Leonin just feel like a grab-bag of existing racial bonuses:

• STR/CON like dwarf, orc, half-orc, orc, goliath, orc, minotaur (maybe they should have just released a Theros orc...)
• extra movement like an elf
• natural weapons like tabaxi, aarakocra, minotaur, lizardfolk
• extra skill like many races
• cat person like tabaxi, shifter

The roar is the only thing that makes them feel distinctive and I feel like all the niches they fit in are already full. With only two new races in the book I would have liked to see something that opens up a little more space with regard to new character concepts.

But to OP's point, I guess they would be pretty good Barbarians in addition to the above.

I agree that the stat blocks are nothing special. I personally hoped for a +1 STR, +1 CON +1 WIS/DEX as the Triton are the only 3 stat boost race that I can think of.

I would also agree about the grab bag opinion, although I feel like you could say that about a lot of races. Most races have some sort of overlap with how you broke down race attributes. You will notice that is a common layout for races in general. The Leonin racial options give it a decent rating in terms of balancing scales. Overall I hoped for something more/different, but I like lions so Ill try it out on a short campaign coming up. Paladin is a good option Conquest or Veng. Perhaps they just don't synergize well with any particular subclass, but are just a good option for most melee.  

So just a thought I’m relatively new to DnD but if you ran a Leonin-Monk  with his natural claws being a d4 on unarmed strikes and same with the monk, at first level wouldn’t you be hitting with 2d4 plus any modifiers? There could be a mechanic that prevents that I’m not aware of but that seems pretty stacked to me at 1st level 

You would not. Natural weaponry does not stack with monk weapon dice.

Think of it this way. Monks, when they use any weapon, get to choose between their monk hit dice and the weapon's natural damage. If a 5th level monk used a rapier, then they'd choose between d6 and d8 damage. A 5th level monk with a dagger chooses between d6 and d4. A 5th level monk does the same thing with natural weapons as they do with rapiers and daggers.

Ah dang that makes sense, thanks for the quick reply 

It would potentially let you inflict slashing damage instead of bludgeoning with your unarmed attacks.  For however useful that actually is.

I'm a big fan of the idea of a Satyr fighter

I've also played around with the idea of a Leonin Cleric

I know neither are optimal but they seem flavorful.

Quote from scatterbraind >>

I'm with 6th here. Leonin just feel like a grab-bag of existing racial bonuses:

• STR/CON like dwarf, orc, half-orc, orc, goliath, orc, minotaur (maybe they should have just released a Theros orc...)
• extra movement like an elf
• natural weapons like tabaxi, aarakocra, minotaur, lizardfolk
• extra skill like many races
• cat person like tabaxi, shifter

The roar is the only thing that makes them feel distinctive and I feel like all the niches they fit in are already full. With only two new races in the book I would have liked to see something that opens up a little more space with regard to new character concepts.

But to OP's point, I guess they would be pretty good Barbarians in addition to the above.

 With the exception of minotaurs, none of the races you've mentioned exist in Theros, so yes they do feel distinctive if you play them in the world where they are supposed to be played.