Ok ill take a shot at this, remember this is all from personal research and experience and should not be the final end all be all. So don't run out and spend money on just my advice.
Are you talking about buying corals online or via LFS. Both are bagged although online they put them in a foam box with heat packs. Locally they usually put them into paper bags once they are in the plastic bag so they don't get direct sunlight.
I take a 5 gallon bucket with me and they gladly fill it with a few gallons of the tank water and place the corals in the bottom in an egg crate frame that i made to support and separate them.
Lets hit acclimation next. If you use the bucket method that I use its simple from here. Simply use an airline and create a gravity flow from your tank to the bucket with a knot in the airline so its a steady drip not a flow. I usually make sure the bucket is no more than 1/2 full to start this that way if I fill it it will be 50 store water 50 mine. I then add them to the tank.
If you do it via just the bags its different. First turn off tank lights, helps if you have a sump as I use to just turn off the sump light and place them in there and float them for 15 minutes which is usually what is needed for the bag and tank temps to sync. After that I use a syringe or a measuring cup and add 1/2 cup of water every 5 minutes, removing water first if I need to. I do this 4 times. After that I add them to the tank.
Now on to placing them into the tank, Turn off the tank lights, moonlights can stay on but be careful if you have other corals in already, some corals will extend tentacles at night and may sting or you can damage the coral itself if you hit them wrong. Anyways, place the corals at the lowest possible, SECURE place in the tank, I don't epoxy them just yet. Sometimes I rubberband them to a lower rock (carefully). Over the next week you want to experiment with different levels and flow areas of the tank to fine where the coral opens up the most and isn't thrashed around. Once you find that use an underwater compatible epoxy. I use mighty puddy. and place it in that spot.
Apply the mighty puddy is a little tricky at times but heres the best way to do it. Take the puddy and mend it blending the two layers together till its one solid color and then let it sit for a few minutes to start to curing process. Then while under water apply the epoxy to the rock, plug, or whatever the coral is attached to. Now shove that into the crack, hole, etc.
Never apply the epoxy directly to the corals as most times it will stick for a little while but the coral will develop a slime layer and remove itself from it. If the coral isn't attached to anything its best to rubberband it to a rock or such and let it attach itself.
Crazy glue is another way of attaching corals to a spot but I have never used it.
The live rock part im not as familiar with, basically what I do with my new LR is put it in a QT tank and wait for the spikes and once its done I add it to my DT, if no spike within 2 weeks I take it as being Cured.
