Unlocking Slots on the wheel
If I understand your question correctly, the answer is for certain slots you will need points only and other slots you will need points and a gem in the vessel.
If you take a look at the top right quadrant below, slot A needs 50 points allocated to unlock both the Dedication and Conviction perk.
If you take a look at slot B, you can unlock the Dedication perk by allocating 75 points, however to get the Conviction perk, you will need to allocate a gem in the vessel on the Gem Atelier tab.

How many points for Stage 1,2,3
STAGE 1: Within a single quadrant, you need 250 points (Level 300)
STAGE 2: Within a single quadrant, you need 500 points (Level 550)
STAGE 3: Within a single quadrant, you need 1000 points (Level 1050)
They points must be within a single quadrant or else it won't count towards the next stage level. For example, you have 1000 points and you have 500 points in the red quadrant and 500 points in the green quadrant. You will be at Stage 2 for both quadrants and not Stage 3 even though 1000 points are allocated on the wheel.
I hope this helps!