Welcome to the 3x3 section of this webpage. This will probably be the most developped section but as you can find 3x3 methods all over the internet, I don't think this will be the most original.
Where to start ?
Generally people start with layer-by-layer methods. Not necessarily the best but the advantage is that most people know them and will be able to help you with them. Also, the transition between layer-by-layer to Fridrich is quite easy (the Fridrich method is the most popular among speedcubers).
Where to continue ?
Generally, people switch directly from layer-by-layer to Fridrich. I think a good continuation good also be the keyhole method (easy to learn, and pretty fast). You can also try other popular methods like Petrus or Roux (at least have a look at them).
What about speed ?
The main methods used for speed in competitions are Fridrich (also called CFOP), Petrus, Roux and Ryan Heise. Just to summarize a lot, I would say: Fridrich is the easiest and the most popular (popular = easy to get tips, tricks, advice and so on), Petrus is more difficult in my opinion but I really like the idea behind it (create a 2x2 cube, then expand it to a 2x2x3, and so on). Roux uses inner slice intensively (you may like it or hate it) and Ryan Heise has the advantage not to require any algorithms to begin with (if you learn it for speed, you'd better learn some algs...).
There surely are a lot of other fast methods out there (sorry if I forgot your favourite one) but my purpose was not to make a full list of them, only to provide people with some options. Also, if one day you are fed up with Fridrich you could try the EOline method. I really enjoy solving it even though I am really slow with it.
What about fun ?
Well, at the beginning, I don't think it is a lot of fun solving the cube. You spend most of the time looking for pieces and really the only satisfaction will be to see it solved at the end. This will change with practice and you will see you are able to have some fun solving a cube, especially with experimental methods like the sandwich method, the human Thistlethwaite method or the Lars Nielsson's method.