Leona is an aggressive champion by her kit's design. If you play Leona, you should expect to be running a kill-lane. She is not designed to go in solo to fight someone with her passive. In lane she can play very aggressive, but your ADC has to be on-board, or you will find yourself feeding, and getting your ADC killed. The ideal scenario for Leona in-lane is to get her ADC chased, then use Zenith Blade to catch the enemy ADC (probably from bush cover). At this point your ADC should immediately turn to shoot and trigger your passive before your stun hits so you can trigger another passive. Your W shouldn't have much longer at this point, enabling even more damage.

Once you have a level and gold advantage (in purchased items, gold is worth nothing if you don't spend it), you can start playing truly aggressive, not having to bait the enemy laners out as much.

It's easy to underestimate how much extra damage you'll get from properly triggering her passive--and you should underestimate it as it's not something you should rely on, but instead take advantage of when you can. Much like catching Draven's spinning axes, you don't want to put yourself in harm's way just to mark an enemy with sunlight, or get your allies to try too hard to trigger it. While you should both make efforts to this effect, you should still play safe and not chase too far. Getting kited is the best way to get killed. Know when to call it good enough. Even if you don't get kills, if you can scare an enemy out of lane at early levels (say level 1) you will have 1 or 2 or sometimes even 3 level advantage when they return to lane.

I like to run with Exhaust and Teleport. Sometimes I swap out Teleport for Flash, with very good reason. If your lane is overextended (as can happen, since Leona tends to lead to aggressive play) you can get ganked easier. When you get ganked, you can hang behind a bit (Leona already runs slow) while your ADC gets away to tower cover. Now you're trapped with (let's say) 3 enemies between you and safety. Now what? If you have flash, my answer is thus: use zenith blade to hit the enemy champion closest to the tower, and then flash closer to the tower. In many instances this will save both your ADC's life (as you were an easier target) and your own. However, smart warding and map awareness is more important, and so I try to play smarter and just use teleport to get back to lane faster when I have to go b. Other decent options include Barrier, Heal, and Clairvoyance (though I don't run with these myself on Leona).

Leona has a slow movement speed. It's a sad truth. But items can fix this!

I like to do one of two things to compensate for her slow move speed affecting her ability to return to lane:

Option 1) use teleport, which is also useful if you need to gank another lane (who should be warding)
Option 2) get Boots of Speed (yes, basic boots) early, say on your first back.

As a support you are NOT going to get champion kills OR creep kills. This kinda sucks, because it means you need gold gen. On the upside, if your ADC isn't so great, it means you can also stay afloat easier. Get a Philosopher's Stone early. The health/mana regen will do more than you think, and the gold gen is esential. I always recommend getting this item on your first back. If you have teleport, you can probably go with tankier items (like a Ruby Crystal or a Chain Vest) and get boots of speed later. I usually build my Ruby Crystal into a Sightstone or my Chain Vest into a Sunfire Cape.

I believe Leona (as a tank) has 3 main jobs, in this order:
1) Pull Aggro
2) CC the entire enemy team, and mark them repeatedly with sunlight
3) Pull Aggro (again)

When teamfights start, if you aren't as fed as you need to be (from assists + gold gen) to survive long enough to get off your combo (to get marks of your passive), sometimes it works out better to have a bruiser dive in first. Then you Ult the entire enemy team after they try to converge on your ally who went in. immediately follow it up with your W-E-Q combo. Assuming your team has enough AoE, you should have just triggered your passive on most of the enemy team about 4 times.

So one problem I've encountered is that Leona doesn't do very much damage on her own, so it can be very challenging to be a threat or finish an enemy off on your own who is already low. This is added to by the fact that Leona has some pretty terrible AP ratios on her abilities. There are a few solutions to this:

1) Build an on-hit item.
A Wit's End or Blade of the Ruined King can make a world of difference in your threat value in the eyes of the enemy team.
2) Play a different champion.
If you don't like an inherently low-damage champion, then maybe this isn't the one for you. Leona is a tank, and has fallen into the role of support rather awkwardly. She gets into the fray and has to survive there or she will just die every teamfight.
3) Build DPS/AD Leona. I have another build I have yet to finish, but it was humorously effective in bot games, and surprisingly effective in Normal PvP games. This works partly due to the fact that Leona's W (Eclipse) makes her instantly tanky if you max it first. More on this in a different build.