Hi, I'll try and TL;DR/quickstart this for you!
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1)
Your character is from the Kha tribe
Your character's surname indicates that she's from the Kha tribe of Xaela.
Basic information on all the major tribes known as of 3.0 (Heavensward) can be found here, but here is the information about the Kha:
Quote:Unlike most of the Xaela, the Kha live on the fringes of the Xaela lands, actively seeking contact with non Auri peoples, introducing many aspects of those cultures into their own.
2)
Your character is probably not from Eorzea + a quick geography lesson
Here is our current world map:
I'm just gonna highlight two parts of it real quick:
The right, bigger red circle is the section of the continent of Othard where most of the Xaela are from - the
Azim Steppe. It's a vast plain where multiple nomadic tribes roam about, trading and looking after livestock, competing for resources and dominance. Mostly Xaela live here.
Nearby to it - to the south - is
Doma, a country which has been occupied by the domineering Garlean Empire for the last two decades. Its residents are mostly Hyur, but the occasional Raen shows up as well.
To the east across the sea is the island of Hingashi, upon which lies
Kugane: Othard's main non-Garlean-aligned international port. To get there from mainland Othard, you have to cross the Ruby Sea, which is infested with pirates.
...and then
way to the left, the smaller red circle is
Limsa Lominsa, where the Arcanists' Guild is.
SO: Your character will probably have needed to make the perilous journey at some stage from the Azim Steppe, to Kugane, and then to Eorzea.
3)
Arcanima is primarily an Eorzean discipline, so your character - on balance of probability - likely learned it after she arrived in Eorzea. Going back and reading your class quest texts is probably your best bet regarding getting the ins and outs of the ACN class, but in short, it's magic created through the visualisation of mathematical geometries.
Have a think about whether you want her hyuran friend to be Eorzean or Doman; either is a possibility. If her friend was Doman, perhaps falling out with him was what prompted her to go travelling in the first place?
4)
Summoner job has some very specific criteria:
a) Your character must be able to read Eorzean, and basically be good at maths
b) Your character must use this ability to learn Arcanima, i.e. the base ACN class, and reach a strong level of proficiency
c) Your character must ALSO survive (body and mind in-tact) an encounter with a weaker Primal such as Ifrit, Titan, or Garuda in their NM or HM iterations
and optional, d) your character may wish to make some connection with an official research organisation, such as the Sons of Saint Coinach, in order to avoid being arrested for dabbling in suspicious-looking magic
The most difficult thing to do here is c). Primals are extremely dangerous, a little bit rare, and have a nasty tendency to perform a process known as "tempering" - a process which wipes the victim's mind clean of almost everything save a desire to worship that primal. Having a gift known as the Echo can prevent you from being tempered, but that's a whole big bag of chips on its own, and one you should only delve into if it sounds interesting to you.
5)
Don't feel the need to press your character into service with a grand company. Grand company RP can be fun, but if it doesn't suit your character, don't sweat it. Most people do their grand company OOCly.
In fact, that goes for the SMN job too - your character can just be a regular ACN if you like, or even develop into a SMN
during RP, instead of it being a backstory element. I find that's always a good way to do elements you're not sure about, since it allows the people you're RPing with to give input about what they think is good.
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UUHH I think that's everything in short, feel free to ask follow-up questions if anything's unclear or if I missed anything? I did my best to keep it simple but I have a tendency to go on sometimes, haha!