You use the XP table to track your overall character level, then choose the class in which to place each level; do not follow XP separately for different classes.
The D & D 3.5 player characters have more than one level: they have a character level, which is a global measure of their experience and power, as well as one or more class levelswho measure how much of their experience and training has been invested in each of their skill sets (represented in the game by character classes).
For a character who has only one class (for example, a character who begins as a fighter and remains a fighter throughout his career), his character level and class level are the same. : At the beginning of the game, they are Level 1 character and Level 1 fighter; after many adventures, they may become a level 20 character and a level 20 fighter.
However, 3.5 characters also have the ability to multiclassing. This means that when you gain a new level of character, you do not have to place it in the class you already have. You can instead place the level in a new class. The characters who take a level in a prestige class are still multiclass characters because you can not take a prestige class at level 1 (that's what differentiates them from basic classes).
To determine when you are going up, use the XP table; This tells you when you gain a new level of character. Each time you gain a new level of character, you choose the class in which to register: When your Level 1 fighter becomes a Level 2 character, you can choose to become a Level 2 fighter or become a Level 2 fighter. multiclass character, like a hunter 1. / Wizard 1.
In the case of your character, trying to become an arcane archer, you ask:
How does a fighter launch 1st level spells?
… and the answer is: do not stay a fighter! Instead, you will pass at least one level in a class that can cast spells (such as Wizard) to meet this requirement. For example, you can stay with Fighter up to level 6, but then, when you reach the 21,000 XP mark and become a level 7 character, go to a level in Wizard, becoming a Fighter 6 / Wizard 1, ready to take your first level of Arcane Archer to level 8.
You can find out more about the multiclass rules in the SRD section on the topic; the same information is also included in the player's manual. Prestige classes are described in more detail in the Dungeon Master Guide.