Fastest way to gather the fields of valor currency?

I want that trust dude! I just bought the game though and am starting at zero. I thought repeating the training would work but it doesnt. It never rewards the currency.

Exchange a Copper AMAN voucher for tabs.

You can only earn tabs from Fields/Grounds of Valor once per game day (~57 minutes). Only the EXP and Prowess rewards repeat during the same game day. Gil and tabs aren't constantly rewarded.

Doing the timed and daily RoE objectives wll give you Copper AMAN Vouchers, which can be exchanged for 1000 units of other currency.

I dont know what those are and I am only level 30.

Activate Records of Eminence in your Quests menu. Once you complete a few of the Tutorial objectives, a lot more of the list opens up to you. One objective (that doesn't count against your limit of 15) auto-flags every 8 hours, and varies based on day of the week and time of day. That automatic objective and the ones in the Daily list, give you Copper AMAN Vouchers when you complete them, which you can trade to the RoE NPC and then spend on various currencies in the game. Valor tabs are one of the currencies.

http://www.reddit.com/r/ffxi/wiki/qol

That has a list of a bunch of the systems they have introduced since the big December 2013 Quality of Life update, with links to the patch notes that explain how to use the new systems that have introduced.

You should know what RoE is because you are pointed to it with a chat message every time you are in town until you start RoE, and it is available at level 1, unlike Trusts being at level 5 to get the first time.

I think those ciphers are pretty cheap, only like 300 tabs each. Completing a Records of Eminence Limited Time Challenge will get you a Copper AMAN Voucher worth 1000.

Like BungieFan said, you should be getting messages in /say telling you where to go when you're in town.

First go to your Quests menu, and there ought to be a listing for Records of Eminences.
Select Objective List.
Select Tutorial.
Select Basics.
Select "First Step Forward".
You'll be given a description of the task you need to do along with the names of the NPCs and their locations. Follow that and it's pretty easy.

Once you do this, you'll have Records of Eminence available. Three times a day, if you're logged on, a message will flash on your screen auto-starting one of the Limited Time Challenges. These are pretty easy and require you to kill 20 monsters of a particular mob family (Lizards, Beasts, Aquans, Vermin, or Plantoids), kill monsters until you get 5000 Experience Points, or kill monsters until you get 3 seals (beastmen seals, kindred seals, kindred crests, high kindred crests, or sacred kindred crests).

Read here for further details.

Anyway, you will probably only need 2 copper vouchers to get all the Trusts you want. Maybe only 1. I don't remember how many there are. You trade the copper vouchers to one of the ROE start NPCs mentioned in the "First Step Forward" objective. He'll store them, and then you'll have the option to Exchange Copper Vouchers. You'll want to exchange for "currencies", which will give you a big long list. Tabs are, oddly, listed as "Valor Points".

If you wish to not use copper vouchers for it, you can grind it the old way. Fields of valor give less rewards than Grounds that I have noticed.
http://wiki.ffxiclopedia.org/wiki/Grounds_Tome
Here is a handy list of different grounds tomes regimens and the rewards for them. There is a range because as you complete regimens over and over, you get more rewards. For instance, if you do the crawlers nest #7 several times over, you'll get 438 tabs on completion (with the game day cap in consideration as posted above though I haven't noticed that before). Your first completion will be only 219 tabs however. Since you get some very potent buffs from prowess on completion, the killing goes quicker as you stay longer too. I grinded 90 to 99 thf by just staying in one area doing the same regimen over and over and over with stronger and stronger buffs. That is probably the best way to grind tabs traditionally.

For just one copper voucher though I believe you can get both trusts as they are 300 each and 1 voucher gets you 1000 tabs, and it would be much faster if that is your only goal.