Hi Felina:

Because the system does not require you to obtain a recipe before you can craft an item, you can attempt any level appropriate synth if you have the necessary materials.

Many people get help with recipe ingredient/ranks from sites such as Yellow Gremlin (http://ffxiv.yg.com) or Zam (http://ffxiv.zam.com). There are also some available through the lodestone

The synthesis screen will show the last 8 recipes you have made for each craft, but you need to do the craft once for it to show up there. The Training Manuals that are purchased with Guild Marks gained through local crafting leves do not contain recipes but instead grant you more skill for doing certain synths.

Hope that helps!