This site is not meant to be a travel guide, so I won’t cover all of the different areas and boroughs within London. Instead, I’ll provide a short answer to a narrower version of that question in an attempt to address a topic that is the subject of many entire books.
The real question that most Arsenal supporters traveling to London for the first time want to know is whether or not it’s important to stay in North London near the Emirates in order to get the true Arsenal experience. While some might disagree with me, I don’t think it is at all. I usually stay in Kensington, near Hyde Park, when I go to London, which is a neighborhood that is nothing like the Highbury/Islington area. North London, especially the part of North London that is home to Arsenal, is a fairly blue collar area that is not home to a lot of museums, fine dining, shopping, and other things that you might expect when visiting London. Especially if you are visiting London for the first time, I’d recommend staying somewhere closer to the center of the city, so that you can visit and experience quintessential London in addition to getting your Arsenal fix.
If you do stay closer to the center of the city, it’s very easy to get to North London on match day. It’s usually no more than a 20-30 minute tube ride to either the Arsenal Station or the Finsbury Park Station (just make sure not to wear your kit or any other Arsenal colors on the tube, which is covered in more detail in the “Should I Be Concerned About Safety?” question). The opposite also applies obviously, in that if you stay in North London it’s a short tube ride to other areas of the city. If you are coming to the city for a week and only plan on spending a day or two of that in North London though, it makes the most sense to me to stay somewhere more centrally located.
Side Note: Other Football Clubs in London
When visiting London, I would absolutely recommend checking out other teams and stadiums. I would specifically recommend going to lower tier matches, be they Championship (Millwall, QPR, Charlton, Brentford, Fulham), League One (Wimbledon), or League Two (Leyton Orient). I’d generally recommend these lower tier matches over going to other Premier League Stadiums (unless you’re going as an Arsenal supporter and have managed to secure an away ticket), and I would specifically recommend Wimbledon and Leyton Orient for having amazing atmospheres in small stadiums. One thing to remember is that when you are visiting another stadium, you are a neutral, NOT an Arsenal supporter. Absolutely do not wear Arsenal colors, do not mention that you are an Arsenal supporter, and be respectful.