Question on Skill Acquiring

How do you know if you purchase / learn a skill with one character will it be available at the trainer for another?

Example- My elementalist has learned the spell Meteor Shower which is not available till after reaching LA. My mesmer arrives later at Post Searing and they could buy it right away. I have tried for other skills with other characters but they are not there?

How do you know? Which will be there and what will not?

When a skill is unlocked, you will be able to buy it from any trainer in the campaign the skill came from. For example, you will be able to get Meteor Shower from Ascalon City, Yaks, Droks, etc. Since you bought the skill for your ele, it therefore unlocked MS on your account, which allowed you to purchase it anywhere in Prophecies.

Once a skill is unlocked, it will be available at any skill trainer of the campaign the skill is tied to. Note that you won't be able to purchase skills that are not tied to your primary or secondary profession even if that skill is unlocked.

...and elite skills must be capped by each character or acquired via an elite tome.

Find a Balthazar Rewards NPC. You might have one in your Guild Hall, or you can go to the Battle Isles and find one there.

If you click on him and go to the skills tab, you will see a list of skills seperated by profession. If the skill says unlocked on the right side of it, then any of your characters on that account can purchase the skill from any skill trainer.

However, if it is a Factions only skill, you have to use a Factions skill trainer. Same for Prophecies and NF. Meteor Shower, for example, is a Prophecies skill, so your Mesmer (to use your example) would not be able to buy MS if he/she were in Kaineng Center.

Another way to check is if you have heroes. Heroes have access to any skill you have unlocked on your account. So just load up the appropriate profession (or change the secondary) and then see if they have the skill. If it isn't there, then you don't have it unlocked.