Based on my understanding, background blur is due to several factors:
1) Aperture chosen which determine the depth of field. The wider the aperture, the narrow the depth of field. Narrow depth of field mean the foreground will be more detailed than the background.
2) Distance between subject and background. This works together with depth of field. If you background is within the depth of field, it will still be detailed as your subject. Therefore if you want background to be blur it have to far from the depth of field.
3) Focal length, a higher focal length will narrow the depth of field even at the same focus length. So even you are on 135mm and uses f6.3, you still can achieve background blur.
4) Sensor size.The bigger the sensor size the easier to get background blur at the same focus length and aperture.
As such, actually, your current kit lens 18-55 can achieve the background blur you want. You sounds that you have not really understand your camera and basic of photography. What mode you uses to take your photos? Try to use A or S or even M mode? You can try to use A mode, change the different aperture on the same scene and take picture. See the effect as such.
Based on your need, you can considered:
1) Sony 55-200 SAM (There are 2 version of this, the newer one is SAM which is better)
2) Sony 18-250 (Guess this all in one zoom will suit you better, cover most of range you need and capture your kids in most situation)
I won't recommend Sony 50mm f1.8, it is a bad focal length on cropped sensor. Sony 35mm f1.8 on cropped sensor will be similar to your normal angle of view.
1) Aperture chosen which determine the depth of field. The wider the aperture, the narrow the depth of field. Narrow depth of field mean the foreground will be more detailed than the background.
2) Distance between subject and background. This works together with depth of field. If you background is within the depth of field, it will still be detailed as your subject. Therefore if you want background to be blur it have to far from the depth of field.
3) Focal length, a higher focal length will narrow the depth of field even at the same focus length. So even you are on 135mm and uses f6.3, you still can achieve background blur.
4) Sensor size.The bigger the sensor size the easier to get background blur at the same focus length and aperture.
As such, actually, your current kit lens 18-55 can achieve the background blur you want. You sounds that you have not really understand your camera and basic of photography. What mode you uses to take your photos? Try to use A or S or even M mode? You can try to use A mode, change the different aperture on the same scene and take picture. See the effect as such.
Based on your need, you can considered:
1) Sony 55-200 SAM (There are 2 version of this, the newer one is SAM which is better)
2) Sony 18-250 (Guess this all in one zoom will suit you better, cover most of range you need and capture your kids in most situation)
I won't recommend Sony 50mm f1.8, it is a bad focal length on cropped sensor. Sony 35mm f1.8 on cropped sensor will be similar to your normal angle of view.
