If you pop off the back plate off your Strat and look in the cavity, you should see the tremolo springs which go from your bridge to a metal bracket that is attached by two screws to the neck end of the body in that cavity.
Generally, the screws on that bracket are not fully screwed in. You can tighten these screws to move the bracket closer to the body. That will increase the tension on the tremolo springs and hence the bridge.
Seeing as you have gone to higher gauge strings, you will have to tighten the tension to compensate for the extra load. Also, if you haven't got all 5 springs in the trem cavity, you may need to add springs - especially if the bracket is already tightened down to the body. (Note: You may only have a 3 spring trem. I have a 5 spring trem on my Strat, but I only use 3 springs).
The other thing that some players do if they have only 3 springs is to angle some of the springs to increase the tension. So instead of
| | | <-- (springs)
They would do:
/ | \
Once again, you need a 5 spring cavity to do this.
All in all though, setting up a Strat trem is a process that requires some experience - so you may want it to initially get set up by a professional - perhaps take some before and after photos so you can learn from that and make future adjustments yourself.