If you aren't looking at end-game rotations and want an "Eorzean's First Rotation" type of thing, your best bet is to think of everything as a "Priority System":

Questions you have to ask:

1) Does your Job have a "increase damage/critical/attack speed/etc for X duration" Global Cooldown (GCD)? Yes, Straight Shot
2) Does your Job have a Damage over Time (DoT) effect to apply? Yes, 3: Venomous Bite, Windbite and Flaming Arrow (Persistent AOE)
3) Does your job have specific combos? No
4) How many enemies are you fighting? (depends)
5) Do you have a filler skill/combo? Yes, Heavy Shot

This means that for your Global CDs, you should make sure Straight Shot buff is up. Then apply all your dots, figure out which order to use based on duration (room for improvement: apply them such that when they wear off, you can re-apply them all without any of them dropping). Then start filling in either your combo[s] (which you don't have) or using your filler attacks. Optimal rotations fiddle with the order of things but for starting out, this is fine.

Now, using your abilities that are off the Global Cooldown (oGCD) is a bit more tricky and varies from Job to Job. The super-simple "not following a rotation" bit basically boils down to "keep everything on cooldown" without interrupting your weapon skills (called clipping). More importantly, anything that buffs damage, you should try to line up as many of your highest damaging abilities while under the effects of Damage Buffs. 2 things to keep in mind:

1) DoT damage is "timestamped" when you initially use it. This means if you turn on a damage buff *before* using a DoT, the increased damage will continue on as long as the DoT is alive. However, this means if you apply a damage buff AFTER the DoT is applied, future ticks of damage will NOT include the increased damage. TL;DR: Apply your highest-damaging DoTs and damaging oGCDs while under the effects of any damage buffs you have.

2) While lining up your cooldowns is important for progression raiding and such, making sure all your cooldowns are being used is more important when you are starting out. Generally speaking, waiting longer than 5-10s for two cooldowns to line up is a waste of time/damage.

With all that said, you can see how a "real" starting rotation for a lvl 50 BRD (single target) comes about (NOTE: I don't really play BRD so this is all second-hand knowledge from Reddit and other sources. Please correct me if I'm wrong!)

1) Pre-Pull (just before and as the tank is walking up to the boss) Foe's Requiem, Quelling Strikes, Battle Voice, Hawk Eye, Blood for Blood, Raging Strikes,
2) Straight Shot (Internal Release + Bloodletter Proc if available)
3) Windbite (Flaming Arrow)
4) Venomous Bite (Barrage)
5) Heavy Shot (Bloodletter Proc or Repelling Strike)
6) Heavy Shot (Blunt Arrow)
6) <Full Priority System, reapply Straight Shot and DoTs as needed, use Bloodletter Procs, Heavy Shot as filler>

Before the pull, you blow most of your damage-buff cooldowns so they apply to all of your DoTs. Then they open up with a Straight Shot (plus more buffing oGCDs) so you are buffed to the gills. Then you apply 1 DoT after the other, throwing in your best damaging cooldowns along the way. Once all those are done, you fill in the gaps with Heavy Shot (your highest damaging attack) and whatever Bloodletter procs you get until you have to reapply your DoTs. From then on, you just use your oGCDs as they come up. Keeping them on cooldown will help you deal as much damage as you can.

-------------------------------------------------

This is a ton of info, I admit but I find if I know *why* I'm supposed to be using the skills in X particular order, I can improvise when I forget the exact order so everything doesn't go down the drain. Hope it helps!

TL;DR Version: Throw on as many buffing abilities just before the boss is pulled, Straight Shot, apply DoT abilities, use Bloodletter Procs whenever they comes up. Always keep Straight Shot up and DoT's ticking.