There are finger staccato, wrist staccato, and arm staccato; hope they are self explanatory, so when playing staccato, don't just staccato, you need to decide which one. Finger is the fastest, but requires practice long-term -- requires technique development. Wrist is medium speed, so you can still go pretty fast. Arm is slowest because the arm is heaviest of the three, and many teachers forbid it, which is a mistake, because speed depends on not only mass, but also amplitude; arm staccato can be made faster by making the motion smaller. To play staccato you don't need much motion. So practice each separately to make sure you can optimize each one. Then increase speed by making the motion smaller. Then to play fast staccato, use mostly finger, but add very small amounts of the other two.
As others have noted, relaxation is always paramount in piano, and especially paramount in soft, fast play, a very difficult technique -- the holy grail of all pianists, so don't set your sights too high at first! Arm weight concept is very useful; to use arm weight, try the gravity drop -- simply let gravity drop the hand onto the keys to play one note, as if your hand is going right through the keyboard and drop under it, except you let the keyboard stop your hand as you stiffen it at the bottom of the keydrop. Your hand will drop freely by gravity only if you are relaxed. At the end, the hand is resting on the keyboard, and you should learn to feel gravity pulling it down, if you are still relaxed. Gravity drop is not the way to play piano; it is just a way to learn relaxation and feel gravity. Why gravity? Because humans evolved, since we were monkeys (or before), with muscle strengths to equal gravity exactly. Therefore, piano builders designed the piano playing forces to equal gravity as much as possible, so if you can feel gravity, the most constant reference possible, you are in the right ball park. The feeling of playing right through the keydrop is important because you must accelerate all the way down; if you start decelerating during the keydrop, the hammer starts to flop around and produces a shallow sound; that's why Steinway designed the "accelerated action" that's how important this is. That's why you need practice: accelerate all the way down, but relax enough to feel gravity -- automatically.
Never use pedal until your technique is satisfactory, even where pedal is required, because you completely lose control of very note, and your technique will immediately stop improving. Practice pedal technique separately at the very end.