1. John, could you please go on reading the paragraph.
The phrasal verb to go on means to keep on doing something, continue doing what you've previously been doing. In the sentence above, the teacher wants John to continue reading the paragraph because probably he stopped doing that, or he just took over from another student.
2. Prices have gone up again, I'm afraid.
The phrasal verb to go up means to become higher, to raise/rise. So if prices go up, it means that the objects in question have become more expensive. The opposite form of this phrasal verb would be to go down, when those things become less expensive, or rather cheaper.
3. You go in through that door over there.
The phrasal verb to go in means to enter a particular place, to be at a particular location that you previously haven't been in. So, in the case above, you are supposed to enter a particular building through a particular door that someone showed you.
4. The temperature went down ten degrees last night. It's now minus five.
The phrasal verb to go down means to become lower, to descend. So, if temperature went up, it means that it became cooler outside, and you'd have to wear warmer clothes. The opposite would be to go up, like in sentence number 2.
5. I'd like the problem to go away.
The phrasal verb to go away means to disappear, to not exist anymore, to leave. So, if you want something or someone to go away, you don't want to see them anymore. If your problem goes away, then you won't have a problem anymore because it will no longer exist.
6. I'll go over the instructions if you like.
The phrasal verb to go over something means that you will read it diligently, you will pay attention to every detail in order to understand what it is about. If you go over the instructions, it means that you are going to read them so as to know what to do.
7. They decided to go back because it started to rain.
The phrasal verb to go back means to return to a previous location, to be at the location that you were before you arrived at your next one. If you decide go outside and it starts raining, then you will probably go back home so as to avoid getting soaked in the rain.