I main support, but I'm not able to play Kalista until she's free.
What champions are good to support Kalista and how should you play/build them?
I'm asking this Question, because Kalista has very unique mechanics:
THE BLACK SPEAR – ACTIVE: Kalista offers a pact to an allied champion. Should they agree, the item is consumed and binds them together, turning the target into a Oathsworn ally for the remainder of the game. Kalista requires an Oathsworn ally to utilize Sentinel's, Soul-Marked and Fate's Call.
Janna excels in supporting teams.
Howling Gale which launches enemies into the air in a line.
Eye of the Storm shields an ally for a determined amount of damage and gives a damage buff, and it is the only shield capable of protecting turrets.
Moonsoon is a teamfight changer ability as it aoe heals your team and blows away enemies in Janna's ability ratio
Tailwind grants 3% move speed buff! 
Karma can contribute with great magic damage bursts. Additionally, she has a single target movement speed buff. Karma is a special support champ in the sense that she does not posses an ultimate ability per se, but her ult, Mantra, gives her a bonus effect on all of her other 3 spells:
Heavenly Wave + 
Mantra = AOE Heal in a cone in front of Karma
Spirit Bond + 
Soul Shield + 
Kayle makes an ally invulnerable to all damage for 2-3 seconds! Additionally, she heals allies and slows enemies, making her a very nice support pick. Unlike Janna and Karma which build defensive and AP items, Kayle builds Hybrid items like Hextech Gunblade and Guinsoo RageBlade. 
Divine Blessing heals up an ally and grants a movement speed bonus.
Intervention grants invulnerability for 2-3 seconds, which can completely alter the course of a team battle if used well. 
Lux is considered a support/mage. She can tear up teams with her chain of abilities, possesses a 2 unit root, AoE slow, and an amazing Ultimate.
Prismatic Barrier shields an ally for damage. 
Morgana is just like Lux in that she is a support/mage. However, she is considered a better support for the shield she grants:
Black Shield is a very unique shield that blocks magic damage and any negative effects: all snares, silences, blinds, etc. This shield scales with 80% of her total ability power. It is a total carry protector and game changer skill.
Sona is the queen of support in my opinion, closely followed by Soraka. All her abilities except for the ultimate give your teammates a buff and she can not be ignored damage wise. 
Hymn of Valor is an aura granting a buff to damage and ability power. 
Aria of Perseverance is an aura that increases armor and magic resist to allies. On cast, it will heal the most wounded ally nearby. The Powerchord bonus decreases total damage output of the target by 20%
Song of Celerity gives a temporary movement speed buff to your allies, in addition to the aura which grants a fixed movement speed bonus! 
Soraka is one of the oldest support champions, and one of the most powerful ones. She has a single target heal plus armor buff and the ability to grant mana to an ally.
Astral Bleesing heals for a decent amount while granting 20-80.
Infuse grants mana to an ally. No other hero can boast an ability like this.
Wish is a global heal. The epitome of an incredible support heal.
Consecration, her passive, increases magic resistance for nearby allies by 16.
Taric can heal and has an Armor Aura, Damage Aura, and a Stun, which makes him a reliable support character. However, one of his downsides is being a melee character unlike many of the others mentioned.
Imbue, if casted on an ally, heals Taric and your ally. If casted on Taric, he heals himself an additional 40%.
Shatter is an aura that increases allies' armor by 10-30 and increases Taric's by double this amount!
Radiance heals Taric by an amount per second while granting an aura that increases attack damage by 15-45 (and double that on Taric). Make sure to use while taking down towers and in team fights!
Zilean is an amazing support champ with the unique ability to revive champions.
Time Warp is a single target Speed Buff/Debuff
Chronoshift revives a fallen enemy with scaling (200% of AP) restored HP and Mana. That's like fighting a 5.5 vs 5 :)
Heightened Learning, his amazing passive, gives all allies an 8% bonus to experience gained!
Best Answer
This is all from theory-crafting the champion so we will need to see what comes out of actual play, however there are two main interactions with the support.
Firstly we have Soul Marked
This generates extra poke each time Kalista and the support both attack the same target
The obvious point here is that a ranged support would be excellent as you can stack together your poke on the same target. For a melee support this particular passive will be less impressive although it will still help on all-ins.
Secondly we have Fate's Call
Fate's Call disables the support's abilities for 4 seconds, so they will probably want to wait until near the end of the 4 seconds before going on. This will place them on top of the enemy team so you now want champions that can take advantage of this. Obvious choices are:
Morgana, this seems like it should be a really strong combination. You have the range to benefit from the poke passive, while at the same time having an ultimate that benefits from being in the middle of the enemy team. Additionally the knock-up will make it harder for them to avoid the stun at the end of the ultimate.
Janna, flash+ulti will throw the enemy team in all directions and some of them into your team to be picked off.
For both of these though the fact that you land at your maximum attack range weakens the combo considerably though, it gets you close but it may not get you close enough. On the other hand however it does make it less "all or nothing".
Leona, she can land her stun at close range and save Zenith Blade to follow anyone who flashes or uses an escape.
Amumu, similar to Leona, in the middle of their team lets you land an awesome ultimate and you can also follow a fleeing enemy with bandage toss.
There are a number of other champions that offer similar abilities too, Thresh's Box, Wukong's ultimate, Alistar's, knock up. Essentially any champion that thrives on being in close to the enemy team and that is able to lay down the pain.
One interesting possibility here is that if you have a melee support then you have already sacrificed the Soul Marked passive for poking (although it will help on an all-in). In that situation it might even make sense to form the bond with something like a Jungling Amumu. Use your ultimate to pull Amumu in then throw him at the enemy laners, he can then Curse of the Sad Mummy and follow any that get away using Bandage Toss.