I have the Garmin Edge 800. It does most of what you are looking for, but fails with respect to budget.
When it is working well, it does a great job, it will plot out routes for you, like a car GPS - I think that as a cyclist, I am more particular about my routes, so I am not always happy with the route that it picks. I can also plot a route on my computer (I use a site called ridewithgps, but there are others), once I have a route plotted, I can download it into my GPS and it will give me turn by turn directions, and beep if I go off course.

It is not perfect - the software/firmware leaves plenty of room for improvement, and the screen should be higher contrast, but it works.