1. I'm new to IELTS reading. Should I start by doing some general tests?
I think that's quite a good idea. The general tests are a bit easier than the academic ones, so you could start with them and practise the techniques I suggest in lessons here on the site.
2. How can I improve my vocabulary and my reading ability?
Basically you need to read a lot, use a dictionary, and keep a vocabulary notebook. Have a look at this lesson.
3. Will it affect my score if I say too much (e.g. a long example in part 3)?
Don't worry about saying too much in parts 2 and 3 of the speaking test; the examiner will stop you if he/she wants to. Just try to give short, direct answers in part 1.
4. Is it ok to write a sentence beginning "This essay will..." in the introduction to task 2?
I prefer to simply answer the question directly (have a look through my lessons to see how I write introductions), but it definitely isn't wrong to write "This essay will...". It won't affect your score either way.
5. What is the difference between could, should and would?
For this kind of question, I'd recommend that you consult a dedicated grammar book or website. This grammar website looks quite good.