Ranking up with the Grand Company
Solution 1:
In general, you can increase your Grand Company rank by spending Grand Company seals to purchase a promotion. If you talk to the Personnel Officer behind the desk at your Grand Company's headquarters, one of the options available should be "Apply for a promotion." The seal cost for a single rank increase starts off low, but gradually increases as you gain rank.
When you attempt to apply for a promotion, the NPC will tell you how many seals are required, and if necessary, what additional tasks must be completed first. At low levels, simply spending seals is all that is required, but as you get into the higher ranks, you'll sometimes be asked to complete certain quests (given by another NPC in your Grand Company headquarters) or complete other tasks before you can earn your next promotion.