One of the annoying things about too many contemporary museums is that, having ditched old-fashioned closely typed descriptive labels and display cases, they often seem to be pitched at the level of a 12-year-old. So it’s refreshing to go to a museum that really is for 12-year-olds – or, at least, babies to 14-year-olds. Three cheers for the Young V&A, formerly the Museum of Childhood. It’s a combination of museum and playground, with an engaging Alice in Wonderland feel to it.

The £13 million refurbishment of the old museum includes letting in roof light to the big central hall with its fine Victorian mosaic floor; the benches around it have jolly foam cushions that small children can play with. That sets the tone: this is designed for children, partly by children.

It is loosely divided by the themes of Play, Imagine and Design. In the Play gallery, there are little ’uns crawling around an area where items are displayed at eye level, if you’re very small.

The mini-museum starts with touchy things – so you get a Dolce & Gabbana fluffy green tabard in a case (just the job for camouflage in grass) and beside it a shaggy surface you can feel for yourself and a tunnel to wriggle through. Across the room, there’s a marble Victorian bust in a case next to a nice smooth surface, so children can see and feel what smooth is. In the sound area you can hear things like falling rain – and there’s a picture by David Hockney of rain on a puddle.

Then there are letters of the alphabet, with assorted artefacts – so, C for cat has a poem by Michael Rosen and two toy cats; F comes with an Egyptian fish tray from 1500-1000 bc and a small stuffed fish from the 1960s; Z for zebras has a teeny one from an old Noah’s Ark. The Swing and Shake section includes a clay Sumerian rattle, c.2300 bc.

You get the picture; it’s not just a collection of vintage toys – though there are plenty of those – but artefacts (about 1,500) from the V&A’s enormous collection. The assumption is that children deserve good art too, although the fine objects are jumbled up with contemporary pieces. In the storymaking section in Places to Go To, the picture suggestions include a Hokusai Wave print and a bright cloud picture by John Constable. It’s like a cabinet of curiosities for children, with digital and interactive elements besides.

It’s not just a museum for looking at stuff. In every section there are activities to try – I especially liked the Self-Portrait section where children capture their own image on a screen and then trace it on paper – and at least for now, lots of paper and pencils are available. The games area has vintage Ludo (yes, please), but also tables set up for anyone to make up their own game. That section includes an enormous Minecraft screen ready to go and a console with joysticks where a couple of young men were pulling the levers.

The adults have a good time too. There’s a very Alice in Wonderland wall with a dog knocker and a little peephole window and a small door; one elderly gentleman was peering inside curiously. On the other side a sloping floor has squares that play with perspective and mirrors that discombobulate the viewer.

The Imagine section, all red plush, is ready for performances, some provided by visiting actors. Nearby, marionettes and harlequins are in cabinets. Best of all is a splendid Venetian carnival theatre with puppet dolls against a Canaletto backdrop.

The dolls’ houses are fabulous, set on an imaginary street – the carpet replicates the streets outside the museum. They include a splendid 19th-century Japanese stage, an 18th-century model mansion with exquisitely costumed dolls and a 1930s Whiteladies art deco house complete with swimming pool, and a party going on inside with lights and music.

Upstairs – via a helter-skelter staircase with a kaleidoscope mirror at the top – a Design section for teenagers showcases materials of the future (fungal lampshades that grow), fun prosthetic arms (made by a British company) and a space for teens to think out designs for themselves; twice a week there’s a designer on site to help. The V&A is a museum for art and design, and you don’t forget it.

The only problem really is that this museum can’t help but fall victim to its own popularity. It’s free, it’s noisy, it’s fun: how is it not going to be turned into an enormous, packed playground – the perfect meeting place for parents (there’s a café) while their children run riot? But as it is a museum for the young, I’d say that’s rather a good fault.