How To Note Violin Tremolo Effect

The "tremolo-like" effect in the beginning, is it played on one string? or two? How best to note it?

You can read this section: http://en.wikipedia.org/wiki/Trill_(music)

Also there's a good explanation here, keep in mind though that it's not certain everyone interprets it the same way:

http://www.pennuto.com/music/jsb_ornm.htm

The Bach trills thing isn't relevant for modern notation.

There is a difference between a trill and a tremolo (or tremolando as the effect is properly called) and consequently a difference in how they are notated. A tremolo involves moving a stringed instrument's bow rapidly back and forth over the string to produce an exciting pulsating effect. This i notated by writing pitches and their rhythms in the usual way and adding small strokes through the stick. Three strokes is usually interpreted as meaning the tremolo should be performed unmeasured as fast as possible, wheras less than three is not a true tremolo but a series of measured notes. Because the bow does not properly grip the string, a solo tremolo cannot quite match the loudest volume and sweetness of tone compared to a normal bowed note. When a whole string section is playing, however, the effect is impressive and is often used to add excitement (although it can become a cliche).

There are two methods which can be used for notating trills. The first is only used for the interval of a major or minor second, and uses the symbol tr~~ over the note, possibly with an accidental to indicate the interval. This is generally assumed to mean the player should perform the oscillation as fast as possible, although it is also possible for the performer to vary the speed if the trilled note is moderately lengthy, often as a solo effect to add excitement and interest to the device.

The second method can be used to notate trills of any interval the player can perform them between. This is done by indicating the two pitches of the trill both using whichever notehead would fill the duration of the trill. This will often result in bars that appear to contain too many beats but it will be clear to the player that the effect is a trill due to the lines drawn between the notes. The advantage of this method is not only that a greater number of possible intervals can be notated but also a more precise speed of the trill depending on how many lines are shown. Three or more is generally interpreted to mean as fast as possible.

Ah yes, the annoying nomenclature of "bowed tremolo" vs. "fingered tremolo" where the latter is simply a trill of an interval larger than a second (usually).

The Bach trills thing isn't relevant for modern notation.

What if you want to write in a baroque style?