Are they new or old strings? What guage are they? I wouldn't go modifying until you have someone look at it. Don't want you ruining your axe.

The tremolo has been known to cause it's fair share of problems . The guys I know have done one of 2 things , if they want to maintain its use they will add springs in the back to give more tension on the bridge . I believe stock is 3 but there are extra hooks so you can add 2 more . Or they just pin th bridge down and make it a hardtail .

Another part of addressing the problem was done on the Eric Johnson strat . Which if you look at the headstock it is at a slightly different angle causing the strings to be straighter going from the nut to the tuners , and also eliminates the need for string trees .

The nut may be playing a part in your problem. When you bend a note the string actually slips thru the nut a little. If the string isn't returning exactly to it's previous position (before the bend) then it will be out of tune. Hence the invention of locking tuners and locking nut.

If you just replaced the strings then what you are experiencing is normal. Should go away for the most part after you've played a bit.

If your tuning is screwed by using your wammy bar one thing you can do to mediate is try to release the bar in the same way every time you use it. I like to life up on mine just slightly after diving a note. Seems to put the strings back in their "pre-bend" position a little more reliably than just easing off the note.

I don't play tremolo any more for this reason. I will eventually be buying a prestige model s model from Ibanez, but it has locking nut and tuners built in. If you love your guitar enough to invest the money then yeah.

it also could be you simply didn't stretch them enough. on average,each string needs to be stretched at least 3 or 4 times before they settle it. if you only stretch them out once or even twice, that may not be enough. and make sure that you have tugged at them at the bridge and above the nut as well...and you say that your trem floats...on a strat?...odd....most of them they flat against the body. anyways you can try adding a spring to the back. that should make it completely flat against the body and will stiffen the trem even more thus eliminating the need for the block of wood while allowing use of the trem should you desire to....putting some pencil lead shavings in the nut grooves can help as well...

Where do I get extra springs?

Fender leaves the bridge on the strats a little high in case you want to use the whammy so I had to tighten it down.

you should be able to get springs at nearly any guitar dealer - if you don't care about being able to pull back your bridge to get those higher pitches then thats definately the way to go - depending on how close it already is, tightening the screws on the bar that the springs are mounted to to increase tension on the existing strings may solve the problem without needing to leave your home

My guitar had an extra spring when I bought it.
Now I have 4 strings on my strat.
And it also gets out of tune when I use whammy, so I quit from using it!

It does depend on about 5 factors.

if the bridge is a vintage trem it should be set so that it is flush with the body, it will need to be ballanced.
also the amount of string windon around the machine head post is critical, I use about 3 turns.
you can rub graphite from a lead pencil into the nut where the string sits in the slot.
you can use vasalene on the string tree to help lubricate the underside where the strings sit.
finaly its a good idea to streach you strings in when you fit them.
you can also use 3 in 1 oil where the string sits in the sadle, this can also help reduce string breakage.

its a bit of a science getting the trem to do what you want and stay intune. If ying and blackmore managed to do it, so can we...