Primary:

Grasp of the Undying

Sion's bread and butter in almost every build. It's just good.

Shield Bash

Now here's something that a lot of Sions probably rarely use in favour of Demolish. Shield Bash makes trading with W even deadlier, if your W survives.

Second Wind

A good rune to recover a little bit of HP that you lost during a trade. It also scales with max HP, so that 4% missing might become pretty substantial late-game.

Revitalize

This rune is one of the most important ones in the build. The free 5% shield/heal power is nice, but what really makes it great is the 10% increased healing/shielding on targets below 40% hp. This stacks multiplicatively with your heal/shield power (redemption) and Spirit Visage's passive. You will notice the effect this has mid to late game.

Secondary:

Cash Back

A great rune to make the build even cheaper.

Cosmic Insight

Another important rune, this will allow you to cast ignite more often and frequently use Redemption's active. It also helps by reducing time between Despair ticks.

I took a long look at Redemption and realized it gives a little of several things that Sion could use early game: Ability haste, health, mana regen. It also gives 10% healing power, meaning it gives you a 10% boost on every W you cast. It may not seem like a lot, but there are many champs who JUST about beat an early game W before Sion can recast it for free damage. Finally, the active effect should not be underestimated: An AoE heal that gives you 200 HP back at the very least AND deals 10% max HP to the enemy.

CS until you hit level 2, using your Q to shove. Once you do, look for quick Grasp trades and possibly W pops on the enemy. It will chunk them pretty hard. Once you hit 3, start looking for stuns if the matchup allows it. If you manage to get a stun in, proc grasp and back off. Stay passive in your trades unless you are absolutely sure the enemy can't just run you down.

Keep this up until the enemy is low enough - usually around level 6 you should have enough pressure to force them away from the minion wave or straight up kill them with a point blank ult + ignite. Around this time you should have enough gold for some Redemption parts. Always buy Forbidden Idol as it gives mana regen and some shield power, plus it is dirt cheap.

Keep up CSing until you have enough gold for Redemption. With Redemption bought, you now have a surprising kill pressure on the enemy laner. Trade until you both are low. Either bait the enemy into engaging into your W or force a fight with a point-blank ult. The moment you are next to the enemy, or a short time before it, you cast Redemption in the area and ignite the enemy. Watch as they try to get out or continue fighting inside of it to take 10% max HP while you heal up in front of their eyes. Your ult, Q and E are great tools to force them to stay inside the area.

Congratulations, that's how you lane as Paramedic Sion. After you get Unending Despair you heal for more than most fighters can deal with, and your items are cheap compared to most fighter items, meaning they'll likely be lagging behind you.

Okay, great. You can win lane by beating them up using a bunch of item, summoner and rune damage. What happens after?

What happens after laning phase is that you become an absolute teamfighting monster. Don't waste your time splitting, Paramedic Sion wants to frontline for the team. Get in front of your teammates or become a threat to the enemy's backline. Your sustain becomes insane thanks to Redemption's heal/shield power combined with Revitalize, and you can keep your team (or yourself) alive for a bit longer with the active. Force the enemies to deal with you instead of your squishies, but it WILL take them far longer to kill you than it would with most other Sion builds. Mainly because of the insane early game you have with this one, but that aside, it is fun to watch your shields become 1/4th of your HP bar when you're below 40%. And to see a resulting Shield Bash destroy an enemy squishy's face.