Flash is a core spell, there's no reason to switch it out for anything else. Exhaust tends to be a strong pick throughout the game, as it gives you a decent damage debuff. I can't say I recommend any other summoner spells, as Soraka does not use Ignite well.
As Soraka can be very squishy early, I prefer to run a combination of health, armor, flat magic resist, mana regeneration, and health regeneration to make her somewhat more durable. Running three mana regen glyphs gives you a bit more mana sustain early on. You can also run armor in quints if you prefer them to health regeneration.
I'm a support player from the North American Server. My mains in ranked are currently Lulu, Karma, Nami, and Taric. I've been playing League of Legends since season one and I've been playing primarily support since late season one/early season two.
Soraka was the first support I ever played (back when Innervating Locket was still an item). I've always liked her and that never really changed, even when riot nerfed her into the ground because old Starcall was super toxic.
Soraka is fairly squishy in lane, this makes her somewhat vulnerable to poke or supports with high catch potential. Soraka also applies very little pressure herself and is regularly forced to rely on allies to do a lot of the work.
-Struggles against good catch potential -Very squishy -Applies very little pressure in lane
Salvation: This gives you a massive movement speed buff that can help you save low health allies who are trying to run towards you. It has the unfortunate effect of doing nothing the rest of the time, but, in the situation where it's active, it tends to be very useful.
Starcall: This skill gives Soraka slightly more harass, a decent slow, and a lot of sustain. Landing it gives her the Rejuvenation buff, which she can give to allies with Astral Infusion.
Astral Infusion: This skill is extremely good for keeping allies sustained. Remember that landing Starcall allows you to use this to give your allies the rejuvenation buff (while retaining it yourself), which makes your healing even more efficient.
Equinox: This ability gives you some ability to control fights, the AoE silence and delayed snare are quite annoying. It has a long cooldown though, so do be careful about timing.
Wish: This skill gives you a decent amount of global pressure when you hit level six. The instant teamwide heal can turn around teamfights and also help to give you an edge when fighting over objectives. Remember that it actually heals 50% more to an ally when they are at 40% of their maximum health or lower.
Currently there are a few ways you can max your skills on Soraka, but I prefer this one. You get a decent mix of sustain for your carry and self sustain. You can also put three ranks in Starcall before maxing Astral Infusion or max Equinox second instead of Starcall if that suits your tastes better.
As Sightstone is core on supports in general, mentioning it seems almost redundant. However, the health it provides is quite beneficial on a squishier support like Soraka. Redemption is a core item on most supports and Soraka really benefits from the health regen and health it provides, plus the passive that increases healing is very good on her. Oracle Alteration is another core support item, that should be gotten fairly soon after hitting level 9, as it allows you to get even more vision control, which is important for keeping an eye on objectives such as Baron Nashor and Dragon.
These are the best defensive purchases for Soraka currently. Mikael's Crucible should be purchased against teams with a lot of hard CC that can be cleansed, plus the passive that converts mana regen into health regen is quite nice for you. Frozen Heart is a good armor purchase that also gives you a lot of CDR. It also gives a lot of armor. The aura is particularly good against AS heavy carries like Vayne. Zz'Rot Portal is a good purchase if your team would benefit from being able to passively push a lane and no one else has bought one. Randuin's Omen is good against champions who are likely to crit a lot ( Jhin, Yasuo, etc), but it's also just a good all around defensive item due to having health to back up its armor. Locket of the Iron Solari brings a massive shield active to the party that has a lot of uses, especially against teams with a lot of burst. Banner of Command pairs well with Zz'Rot Portal and is, in general, good again teams that have very little physical damage to kill the promoted minion with.
Soraka largely doesn't need to purchase offense based items besides Zeke's Harbinger and Ardent Censer. If your carry would benefit from an attack speed buff, then Ardent Censer can be pretty good since the cooldown on Astral Infusion allows you to apply it constantly. If you're on a team with multiple champions who utilize attack speed well you should consider it, since it further buffs your heals and is pretty inexpensive overall. Zeke's Harbinger is particularly good if your carry is already building a standard crit build, since it will bring them to 100% crit chance when the passive activates, but it's good with any carry who benefits from crit chance.
These items are generally situational, but can be useful in or against certain team compositions. Spirit Visage is in a pretty situational (and expensive) purchase for a support, in general. Against AoE or sustained magic damage compositions, Locket of the Iron Solari should be your go-to purchase, but you can get this in addition to that if you really feel starved of MR and health. Warmog's Armor is an item I wouldn't consider in many situations. It's quite expensive and, despite giving a lot of health regen and health, it doesn't give you a whole lot else to work with except for 10% CDR. Banshee's Veil is more of a selfish purchase, but it's somewhat cheaper than Spirit Visage and can be a good backup MR/health/health regen item (in addition to Locket of the Iron Solari) in a pinch. It'a also a good purchase if you have enemies frequently targeting you with hard CC or attempting to burst you down.
Largely Soraka's function in lane is to provide sustain and soft CC. Be sure to use Starcall to get rejuvenation as often as necessary, especially during trades for that extra boost of health regen. Be careful when using Equinox, as its cooldown is quite long early, and be careful to manage your mana well. Use Exhaust when necessary to debuff an enemy's damage and don't use it purely to chase someone down unless you're certain it's going to be worth it.
You will be more able to get rejuvenation more often if you place Starcall properly, so do your best to apply it with Astral Infusion to priority targets. I can't stress this enough: be sure to utilize any item actives you have when necessary, those actives are useless if you never use them. Be watchful of your allies' health bars and use Wish to help turn around the fight if necessary. The same goes for Exhaust as applying that to a high damage target, such as an assassin, when they jump onto a carry can make a lot of difference. Don't forget that you can use Equinox (which has a fairly decent range) to foil engages, disrupt abilities such as Life Form Disintegration Ray and Curtain Call, and to prevent enemies from escaping with Flash or abilities like Arcane Shift.
Remember that Salvation will only give movement speed when you and a low health ally are running towards each other, as this can make a lot of difference when trying to escape from a poor fight.
Rejuvenation works the best when you immediately heal an ally upon hitting Starcall, this allows the health regen to make up for the health cost of Astral Infusion. Rejuvenation gives you (and the first ally healed when you get it from landing Starcall) movement speed when "not moving towards enemies". That last clarification is important to remember, as you can get the movement speed from rejuvenation by moving laterally as well as away from enemies. This can make Starcall and rejuvenation useful kiting tools for you and your carry.
Soraka works well with a lot of carries and team compositions, however champions such as Lucian can utilize the movement speed from rejuvenation quite well. Varus is fairly good at applying pressure on his own and dealing enough damage to make them regret focusing you, which allows you to focus on keeping him sustained in trades. Champions like Ezreal are simply very safe picks that can also dish out a decent amount of damage on their own. There aren't a lot of champions Soraka works badly with, in all honesty, but I can't recommend low damage carries that can do little on their own
Soraka is quite squishy early, which can make dealing with poke difficult. Jhin can deal quite a bit of damage in a short period of time, can do it at a fairly long range, and has pretty good catch potential for an ADC. Blitzcrank is often the bread and butter "counter Soraka" pick, as he exemplifies one of Soraka's biggest weaknesses. Leona also has very good catch potential and Soraka has very little to deal with her if she gets caught herself. In general, Soraka tends to suffer against champions who have strong catch potential and longer range.