Allow me to preface this by saying that this is not coming from personal-playstyle experience, but leading raids and watching / working with the priests in my guild that run Smite spec, I have a good amount of observation.
First and foremost, your most efficient heal is through Atonement procs. Simply spamming Smite is enough to keep most tanks topped through regular damage. Take note of the 15 yard range requirement on Atonement, however. If you're trying to heal up a group/raid member and they're not within 15 yards of the target that you're Smiting, they're not going to get healed. As LessPop_MoreFizz mentions in his comment, you're replacing Heal with Smite, and relying on the 15 yard range rule. It requires a bit more spatial awareness in that you need to know where your targets are as opposed to just yourself. Heal is certainly less effective, but should still be your filler spell (besides Penance) when targets are out of range or you absolutely must heal a certain target right then and there. Also, you should definitely be running with the Divine Accuracy Major Glyph, as that essentially makes it impossible to miss with Smite on any Boss level target.
Secondly, you should note some of the big interactions with your spells:
Penance is great for burst healing and quickly getting Grace stacked on
a target.
Power Word: Shield, as you mentioned, is important due to Rapture procs, and managing these is key to your mana management.
Prayer of Healing remains a strong group heal (not only with just Inner Focus) especially so as Disc, since you auto-proc your Divine Aegis shield off PoH regardless of crit vs. non crit.
Power Infusion is considerably stronger when used on yourself as a throughput cooldown. In the past, it's been reserved exclusively for mage/warlock types, but it's perfectly viable to use it on yourself now. It's practically a "use when up" spell due to it's relatively low cooldown.
Healing your Weakened Soul targets with FH, GH, Heal or Penance results in higher crit chance, not only proccing DA but also Inspiration. Note that this does not include the Atonement heal, so if you're vehement about keeping Inspiration up, watching Weakened Soul is a very good way to help maximize uptime.
Generally speaking, while popping Archangel is great for your mana, note that Evangelism actually increases your damage done by Smite and reduces the cost of both it and Penance by more than Archangel provides overall. If you're looking for instantaneous throughput, it's oftentimes better not to pop Archangel. However, for prolonged throughput and mana effeciency, you definitely should. That choice is obviously reserved for your feel of the current fight and how much effective healing you need.
Binding Heal, Greater Heal and Flash Heal all have their place, though they're certainly a bit more niche as Disc.
Greater Heal is still your largest single target heal, and capable of being spammed, which edges it out above Penance ever so slightly if your tank is taking continuous large hits. That's not to say you shouldn't be mixing them together, but rather that you can't chain Penance back to back.
Flash Heal is still good if someone needs a rapid heal to avoid dying in the next 2 seconds, and you know that they wont survive without a good amount of health in a short amount of time. The raid encounter Chimaeron in Blackwing Depths is a perfect example of fights where this is often utilized.
Binding Heal is your throughput go-to when there's a good amount of damage going out but you're focusing on getting a certain target back up. It's effective in smaller groups like 5 mans, or raid fights where you're supposed to split up. Nefarian phase 2 comes to mind. It's not mana efficient, but it's great for getting quick health to two targets at the same time, and it's overall more efficient than Flash Heal.
Lastly, as an overall word about Cataclysm healing. The healing game early on is a lot more about teaching you how to play triage and get out of the Wrath mindset of "heal everyone to 100% in the next 2 seconds". If a player is at 25% health, with 0 chance of taking damage in the next 10 seconds, but your tank is at 70% and taking constant damage, you pick the tank. Mana regeneration and spell costs for fresh 85 characters is rather punishing, admittedly. If you're having trouble maintaining your mana pool, however, ask yourself these questions:
- Can I get more spirit somehow? Gems, reforging, elixirs, etc.
- Am I using mana potions when necessary?
- Am I using all of my helpful CDs? Pain Suppression, Power Word: Barrier, etc.
- Is my tank using his CDs? Shield Wall / Barkskin / Divine Protection / Icebound Fortitude and any other number of tanking CDs they have.
- Are my DPS taking damage from things they shouldn't?
A special note for the last point there regarding DPS taking damage. As noted, gone are the days where healers can heal through stupid mistakes. If DPS are taking damage to things that they can be moving out of, avoiding, or that they can prevent (like interrupts, using CDs of their own), it's little that you can do to help.
I know some of these last points were more generic than specifically focused on Discipline spec, however I feel that they're important to note since it sounds like there are some general issues in progressing from normals to heroics (and logically, subsequently, to the raid scene) where each "tier" gets progressively stricter in performance, gear and knowledge requirements.
This is somewhat old and written for hunters, but really, the advice it gives is universal.
The short version is, push more buttons. If you're clicking spells, learn to bind them to keys, maximize your GCD's, and always be doing something. The single biggest difference between a bad player and a decent one, is not gear or addons or spell selection, but simple, raw, Global Cooldown usage.
Best Answer
This tends to happen at the end of every expansion, where the people that have been playing longer just want to coast through and set higher and higher standards to get into their groups. It doesn't help that Garrosh on normal is a long fight, so people don't like to wipe on him multiple times. Unfortunately, your options are a bit limited:
Join a guild
The best option is to join a raiding guild that actually raids weekly. Note that SoO has been out for close to a year at this point, so many raiding guilds are experiencing fatigue. It becomes harder and harder to retain a consistent crew, especially as WoD beta invites begin to open up.
Something you can do is to look at guilds you think you might be interested in and see if they are actually inside the raid at the times they claim they raid. So, if a guild says they raid Tuesdays and Wednesdays, /who their guild on Tuesday night and see how many people are online and raiding. You won't be able to distinguish between LFR, Flex, or Normal, but if less than ten people are online, they may struggle to raid normal mode weekly. (Or, maybe they fill in with consistent out-of-guild members.)
Join a cross-realm group
It sounds like you're already trying this and not having much luck. There are addons like oQueue which can help with cross-realm raiding and may let you find some other group that doesn't require a Garrosh kill.
You can also try pleading by showing off your ilvl and your actual normal mode/flex mode experience. I don't recall significant mechanical differences between flex and normal aside from tighter dps/healing requirements, but it's been a long time since I've done Garrosh on flex. Maybe you'll get lucky and someone will waive the achievement requirement for you just to get their run going, especially if you demonstrate knowledge of the mechanics.
Or, you can try to cheat the requirement with an achievement faker or by linking someone else's achievement in the hopes nobody reads the name. However, if you do this, make sure you're prepared for the fight in advance, or else the raid leader will figure it out and be unhappy.
Lead your own raid
I'm sure you're not the only one in this position. You could try forming a group yourself (thus, setting your own requirements) and attempting to find people that are willing to join. Be prepared to spend a while setting up the group, and these groups usually are only good for one night.
If you don't need gear, extend and repeat until you work your way to Garrosh. Some people won't mind skipping the first half of the instance for a shot at the BoA weapons and tier token off Garrosh.