These look good for a first attempt, well done on the shooting stars.
what you can do to help improve things is watch the settings. astrophotography (as that is what you're doing) is about exposure. it looks like you've used a high ISO which has made the pictures a little grainy.
I'd suggest not going above 400ISO, set the camera to an exposure of about 5-8 seconds so that the tracking of the stars doesn't show. manually focus on infinity if you can or on a bright star and then try to set the aperture to 7.1 or slightly higher, this will improve your chances of a crisp shot. in lightroom which i recall you said you had. remove much of the colour saturation, this will reduce the amber flare from streetlights. slide the shadows slider to the right and highlights slider to the left this will show more stars and with a little playing you'll get more defined shooting stars. if you can set the camera on a timer or use a remote so you don't have to touch it that would help too as even on a tripod it'll move which will soften the stars.
as for the moon shot, it looks like you've over exposed this as you can see the dark side of the moon, what you want to get is the light visible with features and the dark side just black. id set your camera up as you would for stars but use a much faster shutter speed, you'll have to play about with it to get it right, also you can use the lowest ISO as the moon is a bright light source. then as before, you can use Lightroom to bring out the definition and remove the sodium light glare. I'd use the moon as your training target as this is the easiest subject in the sky to photograph.
these are good photos and you've picked one of the hardest subjects to photograph because they're so far away. Lightroom will save you a lot of effort and its worth using it on astrophotograhs, might be worth trying another process of your shooting stars pics to see if you can brighten the shooting star trail, as that would be good.
Also you can photograph the ISS as it passes over, even with a bridge camera, if you download a free programme called Stellarium it will tell you what is where in the night sky and also when and where satellites including the ISS are passing over, if its passing over just after dark its clearly visible and you can get a slow shutter pic of its track (like your shooting star pick. if you look on flickr you'll see a lot of people do this, its a very rewarding subject to get and relatively easy once you lens to set up your camera.
If i can help let me know, I've done some astro photography in the past though I'm many a wildlife man.