So we all know that SOME synergy recipes require having a certain rank of skill in some random craft or 3. But what if you dont have this and still wanna skill Synergy?
The answer is the City-Teleport Earrings! They require no skill in any other craft. This is cheap and will mostly cost only Fewell.
You can spend about 11k on getting the rare/ex item and the Silver Earring to get started. Use FFXI Wiki to find where each of them are sold. Each city has their respective rare/ex item found at one of their standard Merchants. Buy the Silver Earring from Lower Jueno at the Gems store.
You can use the Bastok/Windy/Sandy Earrings to start off. These earrings cap at around 19 and you can easily use them to get to 12+. If you happen to actually make the earring, then dont take it since its rare/ex. You will only have to pay for fewell and will eventually make it once you reach a desired level.
Next you will need to take a step up and have your choice of a few Earrings. I personally did Jueno but you can do Rabao, Mhuara or Selbina.
Take any of these to 20+.
Next move on to the Norg Earring. You can get to 30-40 with this.
Viola! You now have some Synergy Skills. Past 40 is a crapshoot. You will now be experienced enough in Synergy to look into some of the higher level armors in order to cap Synergy. These earrings are wonderful, 30 charges and great skillups and they require 0 skill in other crafts.
EDIT:
Tip. Always TRY your hardest to make the earring. That means shoot for the elements, relieve the valve, and repair the machine. If you succeed in the Synth just let the time run out so you can get your mats back. I succeeded on my Jueno Earring several times before I actually took it.