Do you want a zoom or better quality from a prime lens ?
The best 'walk around' lens for the A6000 is the Sony SEL24f18Z
https://www.amazon.co.uk/Sony-SEL24F18Z-AE-E24mm-E-Mount-Camera/dp/B005LLIEO2/ref=sr_1_1?ie=UTF8&qid=1462383376&sr=8-1&keywords=zeiss+24mm
Sharp, sharp and beautiful Zeiss colours. This is one of the best lens you can get for the 'E' mount.
I have this lens for my A6000. It is not stabilized but with a moderate wide angle 24mm = 36mm (in 35mm speak) and great quality wide open from f1.8 you do not need anti-shake
Look around, I picked up a pristine copy for £400.00.
Once you start using this lens you soon forget the 'pain' of paying for it.
I also have the Sony 'E' mount SEL35f18 Sony's 35mm (50mm equiv) this is also a very fine lens and is stabilized and cheaper ! but maybe a little too long for a walk around lens ?
https://www.amazon.co.uk/Sony-SEL35F18-35mm-Fixed-Mount/dp/B0096W1P5W/ref=sr_1_1?ie=UTF8&qid=1462383986&sr=8-1&keywords=sony+sel35f18
I also have the Sony SELP1650 (the kit lens) it is very small, relatively cheap and will do a job but is not in the same class as the other two lens.
The A6000 is capable of superb IQ and is well worth investing in quality.
Have fun.