Bassists are generally divided into fingerstyle players and pick players (and some use both). Fingerstyle players can be divided into players who use 2, 3, 4 fingers... some use more unorthodox techniques and some use their thumbs too. I've never heard of any bassists who use their thumbs solely - I imagine that technique would limit you for speed and probably tone as well.
Examples of Different Techniques:
1 Finger - James Jamerson: This guy is unusual in that he only uses one finger. While it works for him, I imagine it would be quite difficult to build up a technique which uses only one finger. Nevertheless, it is preferable to use as few fingers as possible to maintain an even tone (fingers are of different lengths and thus don't perform exactly uniformly). I use one finger for slower parts for that tone advantage, although two fingers really shouldn't make your tone too uneven so I wouldn't worry about it. Using one finger for everything is probably unrealistic.
2 Finger - Steve Harris (Iron Maiden): Has excellent 2 finger technique and stamina, and can play faster and with better tone than many who use more. Jaco Pastorius is one of the greatest bassists ever, was mighty fast and used only two fingers too.
3 Finger - Steve Di'Giorgio (Death etc): Some people like Di'Giorgio go on to use three fingers, especially if speed is required. There are many variations on the 3 finger technique.
4 Finger - Billy Sheehan: Few players use this technique as it is most often unnecessary. Mainly for virtuosos.
Unorthodox technique: This guy uses his thumb too.
If you choose to play fingerstyle, mastering the 2 finger version is essential and is sufficient for most bassists. Gradually you can incorporate other fingers if you need them. If you choose fingerstyle, use your index and middle fingers and practise alternating them. Start with either one so that you're playing mimimimi or imimimim. You'll get used to it.
If you choose to use a pick, make sure that you're doing it because you prefer the tone and not because it's easier. While using a pick will be easier than fingerstyle, you can ultimately play faster with fingers. Using a pick gives a sharper tone and is often used in a punk setting. Using a pick limits your ability to use other techniques like popping and slapping, and tapping etc.
Most bassists look down on pick players and consider using a pick as an easy way out - but it really depends on what kind of player you are. There are many good bassists who mostly use picks such as Justin Chancellor (Tool), Chris Squire (Yes), Paul McCartney, Andy Rourke (Smiths) and some who use both (but mainly fingerstyle) like Flea, Juan Alderete (Racer X, Mars Volta) and Billy Gould (Faith No More).
I primarily use 2 fingers but I use 3 finger technique for triplets. I'm teaching myself the Steve Di'Giorgio 3 finger technique so that I can play non-triplet stuff with 3 fingers. I'm looking into practising with a pick as well, to add more variety to my technique.