It's a shounen anime where the hot-blooded schoolboy hero (Senku) believes in SCIENCE. He loves chemicals, wires, theorems and making machines.
Then, one day, every human on Earth turns to stone.
3,700 years later, Senku revives. (He knows the date because he was conscious inside his stone shell and counted the seconds.) Senku is a massive freak. The plot's going to need him to know how to make pretty much everything ever created by mankind, so he can't be a normal person. He's a gloating, sneering villain-a-like who'll gleefully cheat, swindle and do any sleazy thing to win in any situation. His friends think he's a scumbag. If he thinks a girl might be attracted to him, he'll be horrified and declare that there's nothing so dumb and illogical as love. At his best (i.e. worst), he's hilarious.
Fortunately, though, he's also determined to resurrect civilisation in this new Stone Age. He's doing it with lots of sneering dialogue and gloating villain faces, but he's also the best friend you could make. If you're sick, he'll manufacture antibiotics. If there's a heavier-than-air gas that can asphyxiate you, he'll make a gas mask. His level and range of knowledge is absurd (especially since it's 3,700 years since he last read the textbooks), but his characterisation's so outrageous that the show gets away with it.
For the most part, the show's simply fun and educational. I enjoyed it and I admire its science-boosting. Perhaps it'll inspire some children to get into science for themselves? It's basically a boys' adventure anime and the characterisation is straightforward, but there's nothing wrong with that.
There are, though, episodes with deeper emotional resonance. Eps.16-17 show us what happened to Senku's dad 3,700 years ago, when the world ended. He was on the International Space Station, so he got to watch it all. He creates hope for the future and leaves something for his son. We also return to him in ep.24. Those episodes were special. I cried (in a good way). There are also simpler but still powerful episodes where he, say, makes a pair of spectacles for a little girl.
The art's deliberately retro. For the male characters, this is awesome and hot-blooded. The female characters, though, barely look human. It's a 20-year-old art style that would be unremarkable in the likes of Kanon, Air and Clannad, but today just looks odd. These girls tend to wear dresses that in real life would stop traffic (e.g. lots of side slits), but here you hardly notice. They're goofy-looking anime characters. "Sexy" isn't in the picture.
One might also wonder about the psychological effects of being trapped conscious in sensory deprivation for 3,700 years. In real life, you can start suffering hallucinations within 15 minutes and you'd expect someone to go insane within days.
As a science anime... well, it's great, but I have a preference for Ascendance of a Bookworm and Maoyu. Dr Stone is roaring with energy and jokes, but its scientific progress feels all over the place. Senku will be developing acid, gunpowder, medicines, gas masks, cola, glass, spectacles and a million other unrelated things. He's constantly jumping from one random thing to another. It's definitely cool and I can see how his world's circumstances would force him to work like this, but he's pulling it all from his memory with at most some trial and error required in the manufacturing techniques. It weakens the sense of scientific progress. Senku's not actually inventing any of this stuff for himself and there's less of a sense of discoveries building on each other. (There is some of the latter, though, e.g. his year-long research to find the formula for reversing human petrification.)
There's nothing I dislike in this show. The art style's weird when it comes to the girls' faces, but hey. The cast are a laugh and I love how the story uses Taiju and Yuzuriha. The story will continue with Season 2 (Stone Wars) and it looks like a blast.