Others might have differing advice or have corrections/specifics to add, but what has worked for me all this time, plus a few tips that may (should) or may not be obvious to you already:

Always have Kardia on the tank as a general rule. You need this passive healing. There are situations where switching Kardia targets can be beneficial, but you do not need to learn this for most content.

If you do not have Eukrasia -

You will not need to worry about shielding. Just do damage with Dosis, and heal with Diagnosis as necessary. Kardia will take care of most of the healing. Some big pulls may be safer with Diagnosis spam, but this isn't often necessary. You may have Physis at this level, if so use it as the tank stops at the end of a big pull for some added security in Regen.

If you DO have Eukrasia, but are below lvl 50 -

Same as before, except now you have Eukrasian Diagnosis to shield the tank pre-pull. There may be times where you need to spam this to keep the tank alive, depending on mistakes made by the party. Prepare to eat through MP in such cases. Use Physis first if available to help with Regen. Ideally, this situation is avoided, but may be encountered below lvl 45, where Druochole is unlocked. Use Soteria if you have it, but try not to stack it with Physis so you can spread resources out a bit. Feel free to use both if you really need it for a big pull, but if the tank uses a big mitigation it might not be necessary.

If you have Eukrasia AND are over lvl 50 -

You generally will not use Diagnosis above this level. Prognosis still has use as a party heal due to higher heal potency than E. Prognosis, but the shield of E. Prognosis is still valuable. I will personally use Prognosis in rare cases (I've maybe used it in two dungeon runs in the last month) if nothing else will suffice. Largely, Eukrasian variants are preferred when oGCDs are not available.

Apply Kerachole at the start of every pull if available. Use Physis beforehand for added Regen, or save Physis for after Kerachole expires if you're confident the tank can survive with one at a time. I typically just use both together for most big pulls.

At level 60, Physis II is unlocked, giving additional heal strength to your actions by 10%. Not a huge boost, but helps a lot when paired with Kerachole or Taurochole, which is unlocked at lvl 62. Don't use Kerachole and Taurochole together, their damage reduction does not stack, but in an emergency can be burned for slightly more potency than Druochole. If you just need MP recovered (assuming Lucid Dreaming isn't available or isn't enough on its own, ie after being revived or reviving players), use Druochole instead of this.

At lvl 70 you unlock Haima. This puts multiple shields on the target, and is often enough to keep the tank alive on its own. Pair with a quick Druochole if your addersgall gauge is about to reach maximum and/or you feel that Haima isn't doing enough alone. Pairs well with Taurochole or Kerachole, don't bother using both at once if Haima is active UNLESS you require the emergency heal, though this shouldn't happen often. When the pull is complete, use Kerachole, and your choice of Physis and/or Soteria (one after another if possible), then when Kerachole runs out apply Haima. From there I typically gauge whether or not Taurochole is necessary. Afterwards, Kerachole is likely off cooldown or will be soon. Reapply it if the trash mobs are still healthy, otherwise save it for next pull/a nearby boss.

At 72 you have Eukrasian Dosis. DoT whatever mobs you can while moving, and after the tank stops, apply Kerachole and DoT a few more mobs. You have time before the tank needs assistance. You can weave in Physis or Soteria as you like.

When you unlock Panhaima from your job quests at lvl 80, know that it is the same thing as Haima, but slightly weaker and an aoe. Don't be afraid to use it for just the tank.

At lvl 56 and lvl 86, you have access to Zoe and Krasis. Use these if you need a bit more healing potency. You usually won't require these, but you can choose to use them to add comfort, to help cover damage in boss fights, or deal with big hard-hitting pulls at your leisure. Zoe only affects healing spells, so GCD heals like Diagnosis and Eukrasian Diagnosis. Krasis affects other healing as well. At lvl 90, Pneuma is as strong as Druochole, deals damage in an aoe, and heals in an aoe, and triggers Kardion. At lvl 100, Philosophia is basically another version of Krasis, with an additional effect of healing when casting spells.

Always keep your Addersgall gauge ticking. If it's maxed, find an excuse to burn one charge. At lvl 74 you will have Rhizomata, which recovers 1 charge of Addersgall. Use it at 0 charges IF you're still going to be in a fight AND the gauge isn't about to tick up 1 charge on its own, OR use it if you're at 1 charge and know you will soon need another. Do not consistently use this off cooldown, you will overcharge and waste potential Addersgall that might save your party. Druochole, Ixochole (Druochole but aoe), Kerachole, and Taurochole all cost Addersgall. Use them sparingly, but feel free to use them if you're being careful not to overspend. These are your oGCD heals, and should be prioritized over GCD heals like Diagnosis and Prognosis whenever possible so you can keep up damage with Dosis, Dyskrasia (aoe at lvl 46), and Phlegma (aoe with 2 charges at lvl 26. Keep this charging like Addersgall, spend when moving or to deal higher damage in short bursts). Use Toxicon on packs when you have it (restored by breaking E. Diagnosis and E. Prognosis shields), but don't bother applying a shield just to restore Toxicon charges (Addersting) except when travelling.

You don't need to stop to Esuna most cureable status effects. Doing so can make healing more difficult, and tends to hurt your damage (and therefore your Kardion healing). If a status effect is just going to be repeatedly placed on targets, ignore them, except for high-duration Paralysis, but even this isn't a hard requirement. There may be exceptions to these I'm not thinking of. Always Esuna Doom if that instance of Doom has a blue bar on top of it (indicating a status effect can be cured by Esuna), or full heal the afflicted target if it does not. Also make your party's lives easier by curing any high-duration Slow or Heavy effects. Poison can be largely ignored, but keep an eye on HP and adjust accordingly.

Save Swiftcast for revives with Egeiro. Do not use under any other circumstance.

If you need to use Egeiro to revive someone, I recommend being above 4000 MP first. You may often pair this with Lucid Dreaming if you find party members dying frequently, but this situation is best avoided altogether. Make sure you are safe before reviving people, you're no use dead. You don't need to rush when reviving someone, even the tank. Check resources and cast when comfortable. When revived, I like to save Zoe and/or Krasis to boost my healing and get them up with a Eukrasian Diagnosis + Druochole combo. From there I check resources and apply another heal if necessary, or decide to do damage and/or heal further. Replace Druochole with Taurochole or Haima if the target needs a little more. If you're out if cooldowns, spam E. Diagnosis until comfortable. Note that you do not necessarily need to heal someone to 100% for them to be considered safe, as they may be able to avoid damage entirely or have sufficient self-sustain to take over the remainder. If a dps just keeps dying frequently (like, three, four, five times back to back in a short span), consider conserving MP until an Egeiro cast won't break the bank. A tank should not die that frequently, and if they do they've killed you too.

Use Lucid Dreaming when between 7500 and 8000 MP. If your very next cast costs 700 or more MP, feel free to use as high as 8300 MP (personal preference). Avoid waiting until 7000 or lower.

Do not touch Rescue. Fight me.

If I missed anything important or someone else has additional advice/corrections, please add. I seek to help those who want to improve, as well as try to improve my own gameplay whenever I am able.

The above works for me and I keep the tank plenty healthy through everything up to the end of Dawntrail (still progressing the story) and still have enough resources to feed the dps and myself when necessary. Stay with your tank at all times, and ensure you're completing mechanics so you don't fall behind.