You also need to work on your Rank in your home city in order to get into the Castle in San'd'oria. I don't know what rank you are, but you should talk to the guard gate to proceed up in rank in your home city. You need to be ready to fight the Rank 3 dragon in order to get into the castle. That is addition to fame.
This
clicky will explain. The fame requirement is in addition to the Ranking.
Yes, the easiest way to get fame is the quest boog mentioned. Of these options, the cabbage is the cheapest way. If you want to farm, you will need about 150 rabbit skins, I believe for the Fame in Sandoria. If you chose to farm for materials rather than the Selbina quest, the easiest quests in Bastok are the Rusty Buckets quest, the Bubbly Bernie quest, and the qudav helm quest.
What quests you do depend on how much gil you have now, and how much you can spend on quests. The quests I mentioned can be farmed by your level, and are fairly easy to complete - they just take time.
Good luck!