Yeah Kefir is usually only slightly thicker than buttermilk, but it can be strained to make it thicker. It doesn't form solid curds like yoghurt though, but has its own particular fractal-y curd structure.
When not in use I just put the jar in the fridge (as long as it's only been at room temp for <24h) and it can stay there for weeks slowly getting more sour. At some point the kefir becomes too sour to be palatable, but will recover just fine after a feeding or two. I also keep backup grains in water in the fridge for years and have revived one of those after 5 years or more! Apparently the grains can also be dried and revived after years as well but I haven't tried that.
There's lots of info about milk Kefir on Dom's Kefir In-Site (straight from the nineties web!):
When not in use I just put the jar in the fridge (as long as it's only been at room temp for <24h) and it can stay there for weeks slowly getting more sour. At some point the kefir becomes too sour to be palatable, but will recover just fine after a feeding or two. I also keep backup grains in water in the fridge for years and have revived one of those after 5 years or more! Apparently the grains can also be dried and revived after years as well but I haven't tried that.
There's lots of info about milk Kefir on Dom's Kefir In-Site (straight from the nineties web!):
http://users.sa.chariot.net.au/~dna/kefirpage.html