This is one of my first multiplayer games and I've been following my advisers' advice for the most part, but I don't seem to be doing all that well. Is their advice actually good, or should I not be following their advice?
In my experience the best approach to using the advice of the AI is to ask yourself why the AI is suggesting a certain unit or building. A worker could mean you should build more tile improvements around your cities. A military unit could mean you're undefended or need a bigger military. Building suggestions could mean you need more production, or maybe the AI just decided you need more culture.
I would recommend deciding upon a victory condition to strive for at the start of the game, and then working towards that goal. When the AI makes suggestions, take them into account and then decide what will work best for your strategy at this time. As others have pointed out, the AI just makes generic recommendations based on what would probably be a good decision right now, but it does not necessarily coincide with the strategy you have in mind.
You can access the advisor panel by using the button in the top right of the screen. You have the Science, Economic, Military, and Foreign advisors. Generally here they just tell you general information that could be useful. Most of their advice on this screen I do not usually use, as most of it you can gather yourself. The most useful information here I find is the military strength of your rivals, which I always consult before making any serious military decisions.
Generally, you might want to listen to the computer, but only to a certain extent. You have to keep in mind that while you may be going for one strategy, the computer is running calculations for a generic, somewhat-decent move. If you're playing multiplayer, remember that people's brains are always more complex than the computer AI, and as the AI isn't trying to interpret their future moves, only the move that would be best for you in that instant. Computer AI is almost always, without fail, defeated by real people in strategy games, so keep that in mind.
The AI advice is 100% completely useless. I turned off all of their suggestions. You should try to understand the game mechanics yourself. If you try to follow their advice you will build a very generic average civilization. To get ahead in Civ you must reach for something important (a tech, or massing certain units, or wonders, gold etc.) and neglect some other things as you reach for what you need to get the advantage. If you play on lower difficulties you can follow the stupid advice and still win.
More importantly though, how fun is it to play if you are just clicking buttons you're told to click?