In reply to The Lemming:
Point the camera in the opposite direction to the moon, so you stand some chance of seeing the Milky Way. Eg line up the field of view with Cassiopeia, which is easy to see in the NE and is aligned with the Milky Way quite nicely.
Lock off the tripod, and try a 20 second exposure. Iso 400 will do. See if it comes out and check the focus on the camera screen by zooming in.
No point doing longer exposures than 20 seconds as the stars will trail. No point going above Iso 400 as the moon will probably over-expose the sky.
It won't come out with much contrast, but if you take many pictures and stack them with deep sky stacker you might be able to stretch the histogram to help with that a bit.
It's not the best of nights with the moon almost full but if the air is very clear where you are then it might be OK, but your combination of mid-summer, northern latitude and full moon all work against you.
You'll see a reasonable chunk of the Milky Way in with a 35mm lens but you should be able to frame Cassiopeia or Cygnus the Swan nicely which contain the Milky Way. The very best images use ultrawide angle lenses or painstakingly constructed mosaics.
Experiment, the exposure time, Iso and shutter speed are not critical and digital film is free. You can see how you're doing on the screen straight away!