Leona is one of my favorites, and now one of my main supports. I build her purely tanky, even during some crazy jungle Leona games.
Although, she can be run top lane as AP or bruiser, bruiser being more popular.
But for support, starting items can be either
or
. It's a matter of preference, but I always run
because the stats are perfect for Leona. When using Relic Shield make sure to use your passive: Spoils of War as often as possible, on melee and canon minions as they give the most gold.
Leona is a tanky initiative support, your job is to protect, peel, engage or disengage for your carries. If you're not used to being the initiator, talk to your jungler or someone else who has a strong engage and let them go first.
Your build will mostly depend on the enemy team. Your core items of course are
and
.
Boots depends on if they have heavy magic or physical damage, so usually choose
or
.
Merc being my favorite as the current meta has a lot of CC champs.
Your other items will depend on the enemy team, if they have a lot of magic damage, build MR. If they have a lot of AD champs, build armor.
MR:
Armor:

If they have an even amount, say, 2 AP, 2AD, 1 tank. I would probably see who's the most fed on their team.
After you get your 3 main items,
, it's generally easy to tell who's ahead on the enemy team, and who will be doing the most damage. Do I want MR against the 6/0 Syndra? or armor against the 1/3/0 Tristana? Look at the enemy team's items as well, that can determine who's going to be doing the most damage.
During team fights, try not to get yourself or carry lost, protect your carries at all cost, especially if they're fed. Use your Face of the Mountain.
For me personally, I sometimes use FOTM on the person who engages team fights, for example, Jarvan. But also use it on your carry if he's out of position or low on health.
As for what to level and max first, I always start Q. Then level 2 I get E, which is all you need to get your combo off.
You can do it QE or EQ if you're quick enough, just make sure you have your Q stun ready when/if your Zenith Blade lands on someone.
Leona, in my opinion, requires communication from your ADC. I've had games where I engage with her, and my ADC just sits back and farms. It's a terrible feeling. ;_;
Certain champs will be harder to stick to. Anyone with a dash or jump pretty much. Ezreal, Vayne, Tristana. Don't even get me started with Sivir/Morgana bot lane. SPELL SHIELDS FOR DAYS.
Level 3 get your W and max your W first (put points in your ult of course), it's the best because it makes you more tanky, and I've ended up stealing a lot of kills with it, hah.
I would do the QEW combo, my ADC would go in, by the time the enemy was low enough, my W would pop, and end up taking the kill.
What to max second is up to you, I generally max Q second.
Runes and Masteries.
This is highly up to preference. I've seen pro players use all different kinds of runes and masteries for Leona, or supports in general.
What I use for runes is the same rune page I use for all my supports, and I love it.
I run 14MR/14Armor/15AP. Pretty simple rune page. I've experimented with others, but I like this one the most.
For masteries I always go 9/0/21 for my pokey AP supports like Sona, Karma, Nami, etc.
BUT for Leona I go 5/4/21. For the longest time I ALWAYS went 0/9/21 on Leona, but after experimenting I realized I really wanted a point in Offense for Expose Weakness. Every time you damage an enemy with a spell, your allies do increased damage to that enemy by 1% for 3 seconds, which synergizes perfectly with Leona's passive.
But as I said, this all a matter of preference.
I was gonna do synergy list but there isn't enough room in the post. :(
ANYONE WITH GAPS CLOSERS AND BURST!
Graves, Corki, Lucian, Quinn, Ezreal, etc.