First of all - welcome to GC.
Now let’s seen if we can help. It’s your floating bridge that’s giving you the trouble here I think. What’s happening is that the string tension is pulling the trem bridge up and as soon as it does this the strings go out of tune against each other, which is exacerbated by the increase in string gauge (regs are 10-46 I think). It depends on whether you want to keep the bridge floating or not. Having it float allows for up bends as well as down. If you want it to do this, I’d take off the back plate, and tighten the screws holding the spring claw about half a turn maybe. Then go back and retune the guitar, which will take time, as each time you tighten one string, the others will lose their tuning as you are experiencing. However, if you bring them up to pitch equally so that they’re there or there abouts, the differences will become less and less until it holds tune across all 6 strings. It takes time and is often irritating, no matter how long you’ve been playing (they will also go out of tune as they settle in too, but that’s part of playing the 6 stringed beast). The other method is to have your bridge set up for down bend only. In this instance, tighten those claw screws down so, that the bridge doesn’t lift when you bring the strings up to pitch, this will lessen the de-tuning effect of the strings on each other, although you will still get a bit, this is easily corrected with a bit of tuning adjustment. However, there is a knock on effect of altering your bridge setting in that it can affect the height of your bridge saddles and the action might feel a bit different in that it’ll be a bit higher if you ‘lock down’ the trem for down bends only. There will be plenty of YouTube videos that will show you how to do all of this. I hope that helps.