So for the uninitiated, I've been making the comparison that RDM is basically 3.X's MCH as a caster, with melee-range burst/finisher. If you don't know 3.X MCH then... welp.
Dualcast is a centralizing trait around RDM's style: every spell with a cast time will generate a Dualcast proc which makes the immediate next cast instant. Note that if you Swiftcast a spell that ordinarily HAS a cast time, Dualcast will NOT activate although Swiftcast is still okay for skipping a proc spell or Jolt. Dualcast will also cancel if you autoattack or use pretty much any other action. Even Sprint. So weave that crap in after every even spell.
As an aside, Vercure can be used to charge a Dualcast proc outside of combat.
The job gauge mechanic is based around the Balance Gauge; this encourages/forces you to spam your 2 primary "combo" chains about equally (if one mana is 30 higher than the other, the gauge will turn that color and the low mana's gain will be halved until it catches up). A bunch of RDM spells generate mana on use; Jolt and Impact are neutral and raise both (as does the aoe spell Scatter), while Veraero/Verstone raise white and Verthunder/Verfire raise black. The first in each pair there (Aero/Thunder) give 11 mana and have a 50% chance of getting you a proc that lets you cast Stone/Fire, which gives 9 mana.
Stone/Fire, Jolt (filler, +3 both mana), and Impact (procs from Jolt, +4 both mana) are all normal cast times and "low" MP cost. Aero/Thunder have long cast times (5s) and slightly higher MP costs.
RDM's cast pattern will pretty much always be:
1) Use procs, or filler if not available.
1a) Verstone or Verfire procs
1b) Impact procs
1c) Jolt
2) Use Veraero or Verthunder, via Dualcast, depending on which mana you need more of.
When both Black and White mana are 80+, it's time to melee burst. Rush with Corps-a-corps, then 123. At this point you can either backstep and use your finisher spell, or use spell and then backstep. Backstep is optional I guess but it's free damage and autoattacks mess with Dualcast anyways. Alternately if you're already in melee range you can use them the other way around. AA damage is extremely minor so don't worry too much about being thrown out of position.
The melee combo generates a spell finisher "combo ender", either Verholy or Verflare. You are free to use either one you want and have a 50% chance of a free Stone/Fire, but if you use the correct one (Holy if black is higher, Flare if white is higher) you get that proc 100%.
Now both the black and white sides of both "combo" sets have equal potency. In fact all 3 proc-based spells have the same potency (270). There's literally no difference other than managing 2 different sets of procs and maintaining equal-ish mana. Procs all last 30 seconds so you're free to hold one that threatens to mess up mana balance.
tl;dr
1. (Stone/Fire) > (Impact) > Jolt. As priority order.
2. Aero/Stone with dualcast
spam until 80 mana, swishy slash, do it again
As far as off-gcds go, Fleche and Contre Sixte are generic attacking ogcds, Acceleration guarantees a proc of Stone/Fire off of Aero/Stone once every 35s, and Manafication doubles current mana values + resets timers on Corps a Corps (gapcloser) and Displacement (backstep). You'd use Manafication when both mana are 40+, obviously. Finally, Embolden is an AoE buff that raises your magic damage and nearby party members' physical damage (10% each), and drops off over its 20s duration (8>6>4>2).
RDM has pretty decent AoE as well. Spam Scatter (100pot), every 30 mana you can throw out Enhanced Moulinet (200, fast gcd). Scatter has a 25% chance of making the next Scatter give 8 mana each instead of 3.
As a minor note, Embolden does not buff your melee WSs at all but it does not seem to buff Fleche and Contre Sixte either. The most optimal time to start it would be in your melee combo (to give melee DPS the physical buff) before using Verflare or Verholy. Pre-68/70 you could just use it wherever I guess.
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Brief aside on mana: while the gauge balancing is a factor, also consider just that every spell generates certain -total- amounts out mana, which can weigh into your decisions on what is or isn't worth casting (and why Jolt is always last priority).
Verthunder/Aero: 11 Black/White
Verfire/stone: 9 Black/White
Impact: 4 Black + 4 White
Jolt, Scatter: 3 Black + 3 White, Scatter having 25% chance to have next cast be 8 Black + 8 White.
Verflare/Verholy: 21 Black/White
-Can be considered a small refund on your 80+80 investment in the melee combo, plus a free proc of the opposite side to balance out your +21.
Note also that (not personally confirmed yet) if you have equal mana going into the Flare/Holy finisher, neither will have 100% chance of giving your respective proc spell. You must have at least 1 point imbalance to get that 100%.
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Overall I'm having fun with job but I can't begin to speculate on its future. RDM/BLM could be a strong caster pair since RDM will appreciate a steady supply of MP, as they lack ANY job-native MP recovery except for the cross-role Lucid Dreaming. RDM does benefit from having Swiftcast every other cast due to Dualcast so it's about as mobile as BRD or MCH in 3.0 were, if not slightly more.