The "cheats" screen in the debug toolbar has a couple of useful tools for this:
The "Hack Gravity" setting lets you adjust the force of gravity from 1% to 1000% of its normal value. However, you can't set it to 0%. Also, if you're doing this on the launch pad, you'll still be inside Kerbin's low atmosphere, which will cause your craft to still be affected by air drag. (Also, if you use this cheat in space, be aware that crossing SOI boundaries with gravity set to anything other than 100% can be very glitchy.)
Perhaps more useful is the "Set Orbit" subscreen, which straight up allows you to teleport your craft to space for testing. This is the method I prefer for testing upper stages. Just launch the stage alone (possibly attached to some supports to hold it up), then use "Set Orbit" to teleport it into space and run your tests there. Or teleport the whole rocket and manually decouple everything except the stage you want to test.
(Tip: The "Set Orbit" screen has some rather conservative limits on how low it lets you set the periapsis of your orbit, and it can also be tricky to find just the right orbit parameters for testing things like aerobraking from a fly-by. Instead, it's often easier to just stick a temporary high-delta-v "helper" stage onto your craft, teleport it to some arbitrary orbit around the target body and then use the helper stage to maneuver the craft to the actual orbit you want before decoupling the extra stage.)
You don't actually need any mods or DLC for this, since the debug toolbar (including both of the "cheats" I mentioned above) is part of stock KSP. The way to activate it depends on which OS (or console) you're using, but the wiki link above describes the various key combinations. (Usually it's F12 plus some modifier keys.)
One mod that I've found useful as a complement to the cheats menu is Vessel Mover, which lets you pick up and move landed craft and gently place them on the ground. This makes it ideal for testing rovers and return stages without having to manually land them every time.
There's also another mod called HyperEdit that implements similar functionality to the stock "cheats" screen but has some additional features on top of that. However, I've personally found the stock cheats in (the latest versions of) stock KSP, optionally combined with Vessel Mover, to be quite sufficient for most craft testing purposes.