There is a screw behind the grip on the frontstrap that when I loosened It it made the trigger lighter. I havnt tried firing it like this for I don't want to mess up the gun. What is that screw for and can it be adjusted?

It's designed to put tension on the spring.It should only be backed out to remove/replace that spring.Some loosen it a bit to lighten the trigger pull,but that often results in inconsistent ignition

The tension screw should always be bottomed out and tight. You can lighten the trigger pull by grinding the screw, or (preferred) by replacing the main spring and rebound spring (internal) with lighter versions. The trigger pull can only be lightened at the risk of unreliable ignition. Replacing the rebound spring requires major disassembly - not for the faint-hearted mechanic. However the action can be smoothed considerably by a skilled gunsmith. S&W has a schedule of charges for this work, and is probably the most reliable agent.

That screw is not an adjustment screw. It's there to hold the tension on the mainspring. You can adjust the trigger pull several ways, and a good place to start is with the different strength mainsprings and/or trigger rebound springs. Changing those springs is relatively inexpensive, and you can change combinations to get a "custom" fit. If those changes don't work to your satisfaction, a competent gunsmith can hone the action. Good luck.

Like others have said, the screw should be tight, it is not an adjustment. If you want lighter DA trigger pull then having a new (and lighter) main spring and rebound spring along with a longer firing pin (helps with any ignition issues from less spring pressure) installed by a competent gun smith is a good way to go. While they're in there they can also polish on the internals to smooth the action as well. Personally I like the way a S&W DA lightens up and trigger feel smooths out naturally after firing a few thousand rounds. If you're interested in working on your own S&W 586 then I recommend:

The S&W Revolver: A Shop Manual Book by Jerry Kuhnhausen

A lot of people loosen the strain screw to get a lighter trigger, then ask why they are getting light primer strikes.

Change the springs, do not lossen the screw. You can do it yourself if you are at all a bit handy. Just follow the instructions in the video below. Also linked is the spring set I used and the spring tool. (Optional) You may also want to purchase a good hollow ground screwdriver for the screw removal. FYI I used the 14lb return spring in both my N frames and it works well. 8.5 lb double action and 3.5 lb single action pulls on both guns.

#178 S&W K,L,N FRAME SPRING KIT | Brownells

S&W REBOUND SLIDE SPRING TOOL | Brownells

How to install a spring kit (trigger job) with Jerry Miculek - YouTube

I have two 686-6's with 3" barrels. To me the SA and DA trigger pull was close but just a little too heavy. I used a
digital caliper and measured the strain screw and then
carefully removed .010". No light primer strikes, but just
lightens the trigger pull just enough that I'm happy with it
without changing springs. And, the more I shoot and dry fire
it ( w/snap-caps) the better it gets!!!