Changing strings on a Floyd Rose bridge guitar needs a bit of care compared to changing strings on your standard, fixed bridge electric guitars:
Change one string at a time to avoid having to tune up for a long time. Even when you take this approach you will still need to tune longer than on any standard fixed bridge guitar, or even a Fullcrum tremolo bridge such as the Fender Stratocaster.
To tune up the strings, use your tuning pegs first, make sure you unlock the locking nut. Once all strings are tuned up with the tuners, fasten locking nut and finetune your strings with the fine tuners at the bridge.
Once you have followed this approach you will find your strings will stay in tune.
Most guitar players will cut the excess bit of string left at the tuning peg. If you are using a Floyd Rose bridge it makes sense not to cut this excess bit of string: Break a string (it usually will break at the bridge) you can unwind the string and refasten at the bridge end. It will save you getting new strings for each moment you break one individual string.
Floyd Rose bridges have their own sound and you do not need to be a dive bomber to appreciate their sound and feel: The tremolo arm does have a lot less flutter than any Fullcrum Strat Tremolo system. You will even notice this when using the tremolo arm in a subtle manner.
Enoy and hope to see you soon again,