A mix of high security clearance, uncommon personal skills, strong feelings ... and maybe a bit of plot convenience.
by the time she confronts the Crystal Gems, she has acquired:
- fighting skills that make her almost the equal of the Crystal Gems fighting together, and maybe stronger than Steven alone;
Her body is exceptionally elastic and bendy, and because of her official role of jester she is highly athletic and able to perform jumps and contortions.
Her perceived betrayal by Pink Diamond gives her a powerful motivation to weaponize these skills, making her a very fast, agile and unpredictable enemy.
The intensity of her negative feelings gives her additional strength (the correlation between personal feelings, either positive or negative ones, and special powers is a common theme along the series).
Also note that, thanks to the weapon she wields, she doesn't need to hit her adversaries particularly hard: a single swooping hit is enough to get rid of the gems. Steven is hit several times because at first he does not want to fight Spinel and he thinks he is quite immune to her attacks (just as he was not affected by destabilizers and prison barriers).
- a special, advanced injector;
- a presumably rare and restricted rejuvenation weapon;
Even if she chose to stand very still for a few millennia, Spinel still had access to the garden warp pad, giving her a quick link to Homeworld and all the systems reachable thence.
Moreover, as part of a Diamond's inner court she was always with her, so she knew (and had access to) everything available to Pink Diamond, including her armory and colonization tools.
Note that the rejuvenator is pink (so, probably Pink's), and would be an extremely appropriate tool for a diamond dealing with a rebellion.
On 2019-10-10, this was confirmed by Rebecca Sugar herself at a book signing in Los Angeles. I couldn't find a video or a precise transcript of the event, but a few social media post talk about it:
Q: How did Spinel get the Injector/other equipment?
A: Spinel was BFFs with Pink and had access to all her stuff and all her security clearances
reddit post by
Q: How did spinel get the injector?
A: She was Pink’s best friend and had access to all of her stuff, (especially to the warp the entire time ;) ), she was able to access it very easily
Rebecca said at the LA signing that as Pink’s best friend, she had access to a whole bunch of things in Pink’s private inventory and knew where EXACTLY to get her deadly war weapons such as the Injector and the Rejuvenator. So she just made a beeline there and headed out immediately
tumblr post by
(I'm going to link a more direct source if I found out one)
- considerable knowledge and tactical cunning (for instance, knowing that the rejuvenator will reset Steven's gem half).
I don't think Spinel's moves where motivated by knowledge and cunning, I'd rather say they were fueled by sorrow and the desire to vent her frustration.
Was she trying to deprive Steven of his powers? Was she trying to get "her" Pink Diamond back? I think she was just trying to hurt them someway.
[Spinel]: What am I doing? Why do I wanna hurt you so bad? I'm supposed to be a friend. I just wanna be a friend.
movie transcript from Steven Universe Wikia
Finally, the short time span from Spinel watching Steven's transmission and her arriving on Earth can be explained by plot opportunity (I'm quite sure there is an appropriate trope).