Progression formulas, like level-ups, weapon upgrades, monster stats by area, etc., tends to be a big jigsaw where any piece can work as long as it's reasonable, but the other pieces need to be carefully fit into place around it.
If you already have the other parts done - for example you've set your character stats by level, and you have your monsters in the game already, then the best way to figure out the power progression of your weapon upgrades would be to estimate how often/quickly the player will be able to upgrade their weapons, and calculate the gap in power required by the weapons to compensate for the difference in monster power vs. the difference in innate character stats, and use those figures to come up with the stats that your weapon should provide.
For example, if the difference in average monster power between the first stage of your game and the third stage is 50%, and you expect the player to level up 5 times during this span, providing a 30% innate boost to stats, and you expect your player to upgrade their weapon once, then the stats on the weapon in its second form, over and above the first form, should provide a 20% boost to your character's TOTAL power (which will absolutely mean more than a 20% boost - it might mean the difference between +15 ATK and +25 ATK, for example).
If you don't already have the other parts done, then you have more freedom to play around with the numbers here, and you can fit the other pieces into place later. Generally you don't want anything too absurd as far as the upgrades go, in order to avoid a balance pendulum that swings too far with a single upgrade. If all of your equipment can be upgraded, something along the lines of +25% to +50% over the previous form is usually good (with lower percentage increases in the final upgrades due to compounding). If only the weapons can be upgraded, +50% to +80% can probably be explored. You say you want about 9 upgrades, but I don't know the length of your game. If your game is very long (30+ hours), you could probably play around with even higher numbers than these, since I expect there will be much bigger gaps in innate character power as well as more stages of your game where monsters will be getting more powerful on average.