I do a great deal of photography and love getting night/astro/star sky shots. I have spent a great deal but for a versatile setup that will do what you want here are, in my opinion, the absolute necesities:
Canon or Nikon DLSR (I run Canon but if you have no lenses yet, I doesn't matter which, they are both ace)
Tripod - don't spend much on your first $50-$100 BUT make sure the head will allow your camera to point straight up
Wide angle lens - this is where you will spend your money. Its worth it as I will explain below
Remote shutter - there are ways around it, but $30 on Ebay, just get one.
Editing software - I recommend Lightroom. This part will make all the difference to seeing the milky way and not
Thats it. So here goes:
Learn to shoot Manual mode ASAP. This is absolutely essential in shooting night sky. Set ISO high but not too high as to reduce quality, set aperture as wide as possible and then just simply play with different exposure times with your remote shutter. The longer the exposure the more you see but the lower the photo quality will become. For a budget setup this will be key in learning for you. The shorter the exposure the better the quality. So high ISO increases the cameras light sensitivity, and wide aperture lets more light in.
Now a wide angle lens does three very important things in night shots:
1. Lets more light in, this helps with what I said above.
2. Makes star motion imperceptible for long exposures. So a 2-3min exposure wont show the stars as a line but a spot. However, this motion can be a good effect if you combine shots over an hour or two. Takes more to explain so will stop..
3. Gives incredible depth by photography more of the sky.
I can only tell you for price, spend as much as you can afford on both the camera and lens but more so with the lens. You will have it almost forever if you take care of it. The camera will last you 5-10 years unless you get hooked and want to upgrade. Spend money here for high ISO quality but honestly, all DSLR's are good these days for what you need.
Look there is a lot to get into but if you get the items I have listed I will help get you taking shots of the night sky that your family and friends wont believe you have taken. My first round of shots on this I had to show proof of shots from the night i was out (all the shots that didn't quite turn out) because people didn't believe I took them or it was real.
PM me if you want more info or if you get the items and want help on settings. I can give you a step by step guide that should make it super easy as long as you are adept enough to work out a few little things yourself along the way.
Joel