To get a steady feel and sound it is important to anchor your hand. The best way of doing this is by placing the thumb on the top of the pick-up (see photo).
Most modern bass guitars have two pick-ups. One close to the fretboard and a second one nearer the bridge. In the early stages of playing it may be easier to place your thumb on the front pick-up as the string tension is not quite as hard as on the back pick-up. The tone will also sound fuller and heavier, and it will also be easier for your right hand fingers to play the strings.
You should also experiment by playing with your thumb resting on the back pick-up, as the sound is clearer and has more attack and bite.
When playing notes, the rest stroke is used. The rest stroke involves the finger picking the string and then coming to rest on the next string. The following photos illustrate the movement of the index finger in playing the rest stroke, i.e., pick the E string and come to rest on the B string.
Alternate between your index and middle fingers. Anchor your thumb on the pick-up and strike the open G string with your index finger in the same fashion as with the rest stroke technique.
As soon as it comes to rest on the next lower string (D string) pick the open G string again, this time using your middle finger. Maintain this alternating movement between the two fingers. Make sure one of the two fingers is always resting on the D string while the other one is picking the G string and vice versa.
Use mostly this alternating finger style as it will give you more possibilities when changing between notes and strings.