Long post!
If you've ever played Lulu support, Karma is fairly similar to that really.
You need to familiarize yourself with using mantra on each ability however. RQ is your best damage/poke spell with a strong slow zone, RW for a fairly strong heal and very long root duration and RE for engaging, disengaging or team-shielding something like a Viktor ult.
Karma is quite strong early on and she tends to fall off a bit later on, this is why I prefer to emphasize building AP/damage on her as that way she remains relevant for longer.
Runes: 9x Magic pen or Hybrid pen marks; 9x flat armour or health seals; for glyphs you can go Ap/scaling AP/magic resist, depending on your preference and the matchup; for Quints you'll want either AP, Flat armour or movespeed, again it's highly preference and matchup based.
Masteries: 9/0/21 or 0/9/21, the utility tree is too good for Karma not to take advantage of. You'll want the extra defence vs All-in lanes, the offensive against poking/ranged ap supports.
Starting items: Always Spellthief's Edge or Ancient Coin with either 4 Health Potions or 3 Health, 1 Mana, always take the warding totem for laning. Coin is great for lanes where you need sustain and when you plan on upgrading to Talisman of Ascension; otherwise go with Spellthief's edge and sell it later on.
Core items: Always take
and
.
Aside from that, it's up to you what to build. You will either want to build with an emphasis on AP or utility, or mix and match the two. AP support is all about improving your base abilities for some nice Q damage and big E shields. Pure support focuses on building utility items for your team largely to the detriment of your basic abilities.
When I play Karma I tend to mix and match utility and AP items based upon the situation. Magic Pen items like Void/Liandry/Sorcerer's shoes amp up your Q damage a lot without requiring much AP, so when you're mixing your build they're the items you'll want to look for to boost your damage.
Banner/Locket are great vs magic DMG, Ardent Censer is useful when you have either a full AD team or many auto attackers, Zeke's to snowball your ADC, Morello's for it's excellent and cheap stats, Rylai's gives your W a 40% slow so it's basically guaranteed to get the root off, Mikael's vs deadly CC (Annie, Lissandra, Veigar), Talisman for engage/disengage. I could go on. You really just build for what you and your team need, and build what's appropriate vs whatever you're facing.
Viable AP items: 








Viable utility items: 







A good mixed build: 






Karma is really a generalist when it comes to her role. She doesn't have specific synergy with certain ADCs and most of her matchups are fairly even. The only enemy supports I would be wary of are Leona/Nautilus. At level 6 they can probably 100-0 you with their ADCs if you get caught, as they can against any squishy ranged support.