Notes that are played together on one bow (slurred) automatically run together smoothly. Cultivating a good legato sound when the bow changes direction requires a lot more practice, however.
The key to achieving this lies in having a flexible bowing hand. As an experiment, try changing bow direction at the frog whilst keeping wrist and fingers entirely rigid. You will find that it is impossible to change bow direction without there being a distinct accent at the moment you change direction. This is because the direction of travel is entirely controlled by the arm, which by itself is not capable of subtle movements. It is the job of the fingers and wrist to smooth out this motion.
Imagine an upbow moving towards the frog and about to change to a downbow. As the arm changes direction, the fingers and wrist remain flexible and relaxed which allows them to continue to move momentarily in the upbow direction, even as the arm changes direction and starts to play a downbow. It is like the difference between jumping on the ground without bending your legs, which leads to an uncomfortable landing, or doing the same on a trampoline, where the springiness of the surface gradually slows your landing before launching you smoothly back into the air.
Once you have understood these fundamentals, playing the following pieces will help you to refine your technique further.
1. Amazing Grace
The tempo of this old hymn can vary a great deal in performance. In order to get the most from your practice, choose the slow accompaniment (see link) and even reduce the tempo further with the slider. Also listen carefully to the video and notice the lack of any 'bump' as the player changes bowing direction. See how closely you can imitate this legato sound. The first three videos below all show cello players, but they should give an impression of the techniques required.
There is a greater feeling of movement here, the tempo being one in a bar. This will help to give some momentum to your bowing, but be careful to keep it smooth, especially on those repeated notes (for example, bars 1 and 2) where you hardly need to hear the changes. Be careful not to mix up phrases marks, the notes in which are bowed, and slurs, which are not.
This quintessential Irish melody will help you to take your legato bowing to the next level, not so much because it always requires the smoothest legato bowing, but because its ebb and flow gives you the opportunity to mix things up a little bit. You will hear, for example, in the video how the player sometimes bows with a beautiful smooth legato, at other times her bowing becomes a little harder. This is especially apparent as she reaches the highest point in the melody, at bar 14. This sensitivity to context, which in turn allows imaginative variation in the bowing arm is one of the keys to becoming a great cellist.