I'm going to try and break down the primary series sequence as far as I've learnt so far with how I do it in my head. This is probably a bit boring but it helps me think about how I think about it, what I know the Sanskrit names of, etc:
Surya Namaskar A:
Mountain pose - uttasana (forward fold) - hands to shins - uttasana - step back into high plank - knees down and lower down to low plank into upward dog - lift to downward dog - look forward and step to the front of the mat - hands to shins - uttasana - bring yourself up to standing - samasthiti
x 3, working up to x 5 - hold the downward dogs for five breaths so you can catch yours
Surya Namaskar B:
Mountain pose - utkatasana - uttasana - hands to shins - uttasana - step back into high plank - knees down and lower down to low plank into upward dog - lift to downward dog - turn left heel and right foot forward into Virabhadrasana A (Warrior 1) - cartwheel hands down and back into high plank - knees down and lower down to low plank into upward dog - lift to downward dog - turn right heels and left foot forward into Virabhadrasana A - cartwheel hands down and back into high plank - knees down and lower down to low plank into upward dog - lift to downward dog - look forward and step to the front of the man - hands to shins - uttasana - bring yourself up to standing - samasthiti
x 3, working up to five - hold the downward dogs for five breaths so you can catch yours
Standing sequence (step to front of mat and samasthiti in-between each posture):
Forward fold with fingers hooked around toes - forward fold with hands under feet - trichinasa (triangle pose) - reverse trichinasa - elbow on shin (knees bent) and arm overhead - reverse this - legs out wide, bend at the hip, hands to floor - come up half way, hands to hips, bend forward - come up halfway, interlink fingers, hands out behind you, bend forward - release, come up half way, down with hands to ankles - gently come up - stand to back of mat, legs straight, reverse namaste hands/hold wrists/elbows, bend forward over the legs - padangusthasana A (hold knee, lift it, balance) - padangusthasana B (gently turn knee out) padangusthasana C (let go of knee, let leg point forward and up) - stand up straight, block on mat in front of you, bring leg across top of thigh, lean forward and bend until hands touch the floor or the block - utkatasana - vinyasa into Warrior 1 - reverse - vinyasa into Warrior 2 - reverse - samasthiti.
And then there's the seated sequence, but I haven't learnt that yet, so I won't write about it until I have a better grasp of it.
Everything needs to be thought about as you go along, tucking under, working on the strength of the inner thighs, focusing on the breath.