How do I approach the analysis of Elgar's first symphony?

Approaching an analysis of a large work such as a symphony can seem daunting at first, but the task can be made easy and enjoyable by setting out what features of the music to look for (such as harmony/ melody/ rhythm/ form/ instrumentation/ texture), by thinking about what the examiner has asked of you. The examiner might specify what particular aspects of the work they want you to focus on, and they might give a specific passage to look at, or ask about the symphony as a whole, in which case you will need to have prepared and memorised specific and significant examples of each aspect/feature. In the case of Elgar's 1st symphony, for example, the 1st and 2nd movements are full of different thematic material, whilst the 3rd movement is a good place to go for instrumentation, texture, string writing but not so good for harmony and tonality since it is more traditional in this aspect and not as significant or innovative. Examples of exam questions could be along the lines of "Discuss Elgar's use of instrumentation and texture in the 3rd movement", or "How does Elgar effectively wind down the ending of the 1st movement?". In the latter example, it is up to you what aspects of the music you choose to explain, but try to include information that is both relevant and interesting, and don't just write a bar-by-bar account as this will waste your limited time! For example, you could argue that the Ab major tonality of the last page of the 1st movement demonstrates a return to the tonic (since the tonic key of the movement is Ab major) and thus suggests resolution, making it a suitable tonal centre for the ending of the movement. Additionally, the 'motto' theme returns briefly (the theme that opens the movement) and so this might indicate that the movement is 'cyclical', meaning it has come full circle both tonally and motivically, helping to 'wind down' the movement. As I previously mentioned, the key to approaching an analysis is to narrow down what you actually need to talk about, by finding the interesting, unusual and significant passages and aspects of the music. You will not know what the exam question is at this point and if you are just beginning to look at the work, I would recommend going through the piece from the beginning, perhaps focusing on a specific aspect as you go through, such as what chords the composer has used, what key is it in here and when/how does it modulate. Or, you can look at a page at a time, taking into account each aspect as you go through. The method you use is up to you, but try to think about what would have been 'typical' for the period the composer is writing in, and how do they break away from tradition, or perhaps you will be required to point out features that fit into tradition, it depends on the exam question and exam board.