This is a champion I'd rather not ban out almost all of the time when I am a team captain but she is really obnoxious to deal with when she has the right support or just straight up gets ahead.
It's hard to deal with her trading and kiting.
My main supports are Sona/Leona/Janna/Bitzcrank
My main ADCs are Ashe/Jynx/Sivir
What would be the ideal picks here, if not here, what would be the better picks to try out against her?
I always pick him than any other support against Kalista. 
is another good pick. Any one with a good engage can rekt Kalista
Ashe. CC chains are the way to go. Just stay out of her range and hit her with an AA every once in a while.
Xin Zhao is good against Kalista
can be a nightmare for Kalista to fight. His passive armor reduces her damage and his stun is targeted so her passive means nothing.
As for ADC...I say Jinx. Her main damage is her Auto attacks that Kalista can't dodge.
Hard CC counters Kalista pretty hard since her passive becomes worthless.
In terms of your supports, Leona is the best option by far followed by Sona imo. Leona provides all the lockdown you should need to at least force her to back if not kill her outright.
In terms of ADCs, Ashe is going to hands down be your best choice. Her permaslow allows you to get more attacks on a Kalista before she hops away and her ult provides the lockdown youd need for a kill. Her Q is also really nice since it provides so much upfront damage whereas Kalista needs to get quite a few AAs on you for a good Rend pop.
