Additionally,

In order to get Company Seals once you have joined a Company, you have to do fates. I have not checked it out for myself, but I have heard you can get them by also doing Dungeons and Guildleves. I don't know how much truth there is to that. As I said, I never checked.

I did not get my Chocobo until I was level 24. It took me that long to get enough Company Seals to afford it. While my method might be slower, I just did Fates that were near quests I was doing, or on the way to one.