I'm a DM and I have been playing for three years. My party has just now begun looking for mounts and I want cool and unique Ideas. 

Steeds:

Riding Horse:

Hp: 13

Ac: 10

Speed: 60 ft

Hooves: ( 2d4+3) damage +2 to hit

Riding Buffalo:

Hp:23

Ac 12

Speed 45 feet

Charge: If the buffalo moves at least 20 feet straight toward a target and then hits it with a gore attack on the same turn the target takes an an extra (2d6) piercing damage.

Gore: +6 to hit (1d6+4)

Riding Toad:

Hp: 13

Ac 10

Speed: 40

Swimming Speed: 40

Jump: May leap half their movement either vertically or horizontally

Sea Horse:

HP:13

Ac: 10

Swimming Speed: 60

Pouch: Can carry a third medium creature

Elk:

Hp: 13

Ac: 10

Speed: 60

Horns: attack (1d6+2) +2 to hit

Riding Turtle:

Hp: 20

Ac 18

Speed 35

Swim Speed: 40

Bite: (1d4+2) +2 to hit

Riding Snail:

Hp 20

Ac 18

Speed 30

Climbing Speed 30

Bite (1d4+1) +1 to hit

This is what I have so far and I'm looking for cool real world beasts or Fantasy beasts. 

Lion

Ostrich

Camel

Giant Birds

I'm also looking for cool abilities or unique traits for these creatures. Thank you for the help, and please take these for your campaign.

I don't know about anyone else, but personally I would ADORE a pet owlbear. Haha! 

Ohh, give them something like a Tarrasque or a Purple Worm as a mount... but make them capture and tame it before they can ride on it! You could make this be a sidequest that runs through the entire campaign, those are always fun! :)

Quote from Zillion >>

Ohh, give them something like a Tarrasque or a Purple Worm as a mount... but make them capture and tame it before they can ride on it! You could make this be a sidequest that runs through the entire campaign, those are always fun! :)

I would ask them to multiclass into a wyrmling rider( found on homebrew 5e page) then make them work to get a wyrmling egg and allow them to raise it.(wait to allow it to be a mount until the wyrmling is a month old at the very least.)

Giant Goat, Giant Goat, GIANT GOAT

A warhorse is appropriate for paladines, fighters, and war clerics.

Pegassi and Griffins are always inspiring. Unicorns don't work for me.

Of course, the Hobbit showed us it can be fun to ride a ram or a moose, or a giant eagle.

Maybe a nazgul, wyverin or dragon. A flying carpet isn't a mount but it can get you around. Or a flying broom.

Have a field of poisonous flowers that put people to sleep for 1 year if you get within 10 feet of it.  Give the party giant slugs as the only available mount.  The trick is that the party has to figure out a way to incentivize the giant slug to go the direction that the party wants it to go:  through the field.  

Giant Centipedes. Finally, a mount with back seats!