Sequence for starting beginners
Establishing the Fundamentals
1) Begin by establishing correct posture and instrument position.
2) Learn open string songs by ear that can be played pizzicato with both left and right hands.
3) Learn how to hold the bow through the sequence outlined on this website.
4) While learning to hold the bow, separately learn to put fingers down and establish left hand position.
5) Learn to play pizzicato songs by ear while putting fingers down. Separately learn to play open string rhythms with the bow.
6) Learn to play songs with the bow by ear while putting fingers down.
7) Learn a repertoire of simple songs by ear, while separately learning to read music away from the instrument.
Only when instrument position, left hand position, bow hold and basic bow stroke are well established and the ear is developing well should reading music be introduced while playing the instrument.
Establishing Left Hand Finger Placement
Historically, the most common approach is to begin from the open strings in first position, and add one finger at a time, until all the fingers are placed. In group methods, the first string introduced is usually a string that is a middle string for all the instruments - the D string. Once the placement of the fingers has been learned on one string, the other strings are introduced:
As you can see, the problem for the group teacher is that to stay in first position the Double Basses must move to the next string to play the note "G". The cellos must move to the next string for the "A." If you examine beginning classroom method books you will find that the authors sometimes delay the placement of the violin and viola 4th finger in the book, so the violins, violas, and cellos can all play the A open. The double bass is sometimes presented with the open A string (a P4 lower in pitch than the D) rather than playing the A with 1st finger on the G string.
Study the book carefully before you buy it. Some method books make great compromises in logical presentation for the sake of uniformity. By definition, every classroom method book is a compromise to some degree. If you examine the method books written for the specific instruments of the string family, you will find they vary considerably from classroom methods. Consider the double bass. In the traditional Simandl method for Double Bass, the pitches of the so-called "half position" are presented first, and then the player is introduced to the F and Bb scales with exercises. Then the first position is introduced, and the player learns the G scale and exercises. This makes sense if you consider the fact that the F, Bb, and G scales are the only scales that can be played on the bass (using the Simandl fingering) without shifting. Newer approaches to beginning bass pedagogy, such as George Vance's Progressive Repertoire, start with the hand in the "neck block" position. This approach works well for many young students, because it is more comfortable for the left hand. The low positions on the bass require the most stretching of the hand and can cause fatigue and pain for many beginners.
The creative classroom teacher will synthesize exercises from the more idiomatic instrument methods with classroom methods, and hopefully add their own knowledge and experience to the mix so their students receive a logical and sequential presentation. As an old professor said to me, "There are many roads to get downtown. You need to take the road that works best for you."
Starting with all the fingers down
Some teachers advocate the following method: Instead of starting from the open strings and adding one finger at a time, start with all 4 fingers down on the strings. This will develop a correct hand position from the beginning, since it will be impossible to play 4th finger in tune if the hand is not shaped properly. Doris Gazda's method book "Spotlight on Strings" takes this type of approach. Robert Culver also advocates this approach in his pedagogy classes.