Unlocking Sidequests

I played around with characters starting at the different cities and have noticed something unexpected.
Sidequests for main class are not always necessary but they do provide pieces of armor.

Sidequests are abundant for main class areas but when starting a different class on a different city those sidequests are not available.

For example, I started this character as ARC but when I unlocked GLD in Uldah there were no sidequests available in the surrounding Uldah area, but they were there when I created a character with GLD as starting class.

Is there a way to unlock these sidequests?

That's because sidequests are not repeatable. Plus a lot of quests only become available as you progress through the main scenario. Really, quests are only there to help you level your first class/Job to 50, and after that you need to use alternative means of levelling (levequests, Hunting Log, Challenge Log, dungeons/trials, Guildhests, FATEs), not to mention when you level a second class you will gain a bonus to exp received up to the level of your first class (for instance, if your 'first' or highest levelled class was at level 45 any other class you level will gain an exp bonus up to level 45).

Then there is the separate 'rested exp bonus' for idling in a sanctuary, the exp bonus buff you can have active while a member of a Free Company, and even food gives, or at least used to give a small exp bonus too. All these are set up to help you level other classes faster after your first class to compensate for the lack of sidequests by that stage in the game.

There are special repeatable quests though (levequests, beastmen dailies etc), but these are not the 'standard' style of quests and require story completion to unlock (and, in the case of the beastmen quests, a high level).

There are a few low level sidequests in each area that you should be able to take regardless of where you started, but yes, a lot of others are only unlocked in conjunction with the Main Scenario quests in that area (which, for levels 1-15, you'll only get if you started in that area). Since after level 15, the Main Scenario quests are merged into a single storyline, it also means that from that point on, you'll be unlocking all the sidequests as well. It's just the first 15 levels worth that you won't unlock everything, if you're not doing that region's MSQ.

You won't be able to level or gear more than one or maybe two classes with questing, though. Beyond that, you'll need to find other sources for both XP and gear. As Enkidoh mentioned, there are enough XP boosts available that the actual leveling isn't as hard as that sounds. As far as the gear provided by quests, it's the same as is offered by NPC vendors and the same as Normal Quality results from crafting, so you can find equivalents from either vendors or the Market Board.