The Royal Rekka is a multi-part move not unlike Base Goku's old Dragon Fist.
It has a starter attack (a gut punch, as well as a multi-punch EX version) and from that, various paths you can take. Let's break it down:
STARTERS
LP has the fastest start-up but the weakest hitstun on impact. Just enough hitstun to link into the MP Path, not the LP or SP ones.
MP has a medium start-up and a decent hitstun on impact, allowing you to link into a Light attack.
SP has the slowest start-up but the longest hitstun on impact, allowing you to link into a Light or Medium attack.
You can go into the next attack even if the previous one didn't end, skipping it in a way.
The EX version has the start-up time and hitstun of the MP version but gains quite some range, traveling forward at least 2/5 of the screen. It also does more damage by automatically following into a couple more punches. You can cancel out of the final punch into a Path quicker than any of the Normal versions, almost immediately upon impact.
However you cannot go into a Path if the move whiffs. When it's blocked, Vegeta will keep throwing out the additional punches. The hitstun is inbetween that of the LP and MP versions.
EX starter also has the special order of starter> path> ender> path> starter; the second starter in it causes a hard knockdown but if you make that second starter another EX, it will restand the enemy.
PATHS. The stronger the Starter leading up to it, the stronger the Path.
LP hits low. Due to its medium-long start-up speed it does NOT combo off of the LP Starter. Doing it out of MP Starter does enough hitstun for you to follow with a Light Attack. Out of a SP Starter, Mediums will also connect.
MP hits mid. Doing it out of MP Starter does enough hitstun for you to follow with a Light Attack. Out of a SP Starter which does even more hitstun, Mediums will also connect.
SP hits high. Due to its long start-up speed it does NOT combo off of the LP Starter. It also has the longest recovery time. Doing it out of SP Starter causes enough hitstun for you to follow with a Light Attack. Not Mediums.
ENDERS. As with the Path, the stronger the Starter leading up to it is, the stronger the Ender will be.
LP hits low and pop-trips the enemy (similar to N/M Vegeta's Dashing Slide Kick). Out of a Strong Starter, it will flip KO the opponent up for even better juggling options.
MP hits mid and knocks the enemy away. However in the corner, you can catch p2 with a sneaky cr.LK. Doing it out of an LP Path does a simple knock-away, while out of a MP Path results in a Twist KO. Out of SP Path it also does a Twist KO but makes p2 fly up and away higher and wallsplat even harder, allowing for new follow-up options.
SP hits high and launches p2, enabling you to Pursuit and aircombo. Out of a LP Starter it does quite a small launch but it still allows for a persuit into aircombo. Out of a MP Starter it does the common launcher, and out of an SP Starter it will Flip KO the opponent high into the air.
EX Ender creates a small multi-hitting Atomic Bomb, enabling Vegeta to retreat while leaving behind a present for the opponent.