The Moomin books were written over several decades, and the style gradually evolves, from straightforward, if whimsically strange and occasionally wise, adventure tales early on; through more experimentally witty and psychologically complex stories in the middle books; to the melancholy, largely interior narratives of the late stories. Eventually Jansson entirely abandoned Moominvalley for adult mainstream fiction. Comet is from the early period, and Moominsummer Madness (my personal favorite, I think) is a middle book.
Finn Family Moomintroll is the last of what I'd call the early books. The generic-sounding title comes from the English-language publisher: it was the first of the Moomin books to be translated, and it was sold in English-speaking countries as the introduction to the series, though it's actually the third book.
Its original title was The Magician's Hat, which better describes what it's about, though it is quite episodic and only loosely plotted. I suspect the episodic structure may have been what motivated the decision to market it as an intro; it's as good of one as any.
Moomintroll and Sniff discover a top hat with strange magical powers, apparently left by a fearsome cosmic character called the Hobgoblin (I think in the Swedish he was just "the Magician"). The first section of the book is a series of vignettes in which the hat works odd and temporary transformations on anything put into it: trash into little floating clouds, and Moomintroll himself into an unfamiliar jug-eared creature.
The middle section is a digression, leaving the hat behind, in which everyone takes a trip to an island hosting a congregation of Hattifatteners: mysterious, small, ghostlike creatures who consume electricity and spend their lives aimlessly wandering the ocean in ships. There is some conflict with the Hattifatteners (the Hemulen steals their sacred barometer), a fearsome storm, and a lucrative treasure hunt. The Snork Maiden becomes jealous of Moomintroll's doting over a ship's figurehead, a beautiful wooden lady, that he finds on the beach.
Returning home, the Moomins catch a gigantic fish, and discover that the magic hat has transformed the interior of their house into a jungle. But the last section changes the story again by introducing Thingumy and Bob, two tiny and larcenous characters who speak in spoonerisms (in the Swedish they use more of a pig-Latin-like transformation), and have somewhere obtained a precious jewel called the King's Ruby. The Ruby is the very thing that the Hobgoblin has been searching the universe for, and Jansson describes it in ecstatic terms as lighting everything up with ever-changing red fire when uncovered.
It turns out they stole it from the Groke, a recurring figure in the Moomin universe: a terrifying creature who freezes the very ground she walks on. The Groke comes to demand it back, and the family hold a charmingly mixed-up trial in the garden to resolve the situation; but later the Hobgoblin himself shows up to claim the King's Ruby, and turns out to be quite a nice chap when you get to know him. The Hobgoblin has magical wish-granting abilities that he can only use for the benefit of others; this causes some trouble, when the Snork Maiden decides to make her eyes look like the wooden lady's, but it's also used to make everything right in the end.
Finn Family Moomintroll may be the most lightweight of the Moomin books, but it's a lot of fun and kids will like it. As in Comet, the gender roles are still quite conventional; it's only toward the end of the next book that things start to change a bit.
That next book is Moominpappa's Memoirs, easily Jorie's favorite of the series, so we have read it many times. I never read this one when I was a kid, and more's the pity, because it's the silliest and wildest of all the Moomin books. It's also a big jump up in narrative sophistication, playing clever games with metafiction, jumps back and forth in time and an unreliable narrator. I consider it the beginning of Jansson's middle period.
The book has a frame story in which Moominpappa, stricken with a cold, is writing his autobiography, and intermittently reading it to Moomintroll, Snufkin and Sniff, who provide sometimes pointed criticism and commentary. The rest of it is the autobiography itself, which turns out to be a series of bizarre and outlandish adventures, narrated by Moominpappa in hilariously pompous and long-winded fashion.
Young Moominpappa escapes from an orphanage run by a tyrannical Hemulen, convinced by his boundless ego that he is destined for great things. He makes the acquaintance of Hodgkins, a taciturn, engineering-minded creature (he reminds me of Ferb of "Phineas and Ferb") who is building a fantastical riverboat. Sooner or later they run into Snufkin's father the Joxter, who is basically a lazier, sleepier version of Snufkin; and Sniff's father the Muddler, who lives in a Maxwell House coffee can and makes Sniff look brave and competent. The boat is named the "Oshun Oxtra", because Hodgkins suggested "Ocean Orchestra" but the Muddler can't spell.
The gang trick an enormous Godzilla-like creature named Edward the Booble into launching their boat by causing a tsunami with his butt (I always hurt my throat doing Edward's thunderous bellow), spend some time flying through the air with the aid of a strangely animate cloud, and have a run-in with the Hemulen's Aunt, who tries to impose order on the little band but fortunately gets carried off by a swarm of Niblings, slimy-footed creatures who compulsively chew on everything.
Midway through, the book takes an interesting, unexpected turn. The masculine bent of the story (in which female characters are mostly obstacles) is broken by the Mymble's Daughter, a puckish, fiercely independent soul who will become a major character in the series. From this point on, major female characters in the Moominverse will start to increase in number and variety, though it's only fitfully visible here.
Shortly after, they all end up in a country ruled by a whimsical Autocrat, a king fond of amusement parks, parties and practical jokes on what he affectionately calls his "useless subjects". Moominpappa, convinced of his destiny as an adventurer, wants to keep voyaging out to sea and visiting strange lands, but that's not what happens. Hodgkins has all sorts of ideas for modifying and rebuilding his boat, and the Autocrat gives him a job offer that involves the opportunity to do that (on his "20 percent time", as it were).
So instead of venturing on, they all found what is known as the "Royal Outlaw Colony". But Moominpappa is dissatisfied, a harbinger of midlife crises to come in later books: he doesn't think all this is worthy of his greatness, and there's a splendid, both sad and funny section in which he wallows theatrically in self-pity and finds it's something he can actively savor.
And then he discovers a ghost! We flash back to the present, and Moomintroll complains that, though the ghost is a great touch, that bit with the self-pity went on way, way too long. But Moominmamma assures Moominpappa that it's fine... the kids just aren't mature enough to get it.
The next chapter, maybe to balance out the heavier stuff, is a screamingly funny bit in which the ghost, whose dialogue is all overcooked ranting about "the forgotten bones", turns out to be utterly incompetent at scaring people, but quite effective at annoying them. The gang eventually befriend him, with the suggestion that Hodgkins can tell him some frightening tricks he knows (of a piece with the practical jokes he thinks up for the Autocrat).
At one of the riotous all-night parties that Jansson so loves to describe, held in the Autocrat's Garden of Surprises, they meet the Mymble herself (a jolly creature who the Joxter immediately takes a shine to) and her vast family of little Mymbles, the smallest of which is possibly Jansson's greatest breakout character, the uncontrollable and morbidly imaginative Little My. Her role in this book is relatively small but she's already causing trouble. Meanwhile, Hodgkins has been rebuilding the Oshun Oxtra into an even more fantastical flying submarine, and everyone goes for a phantasmagoric and dangerous ride.
The rest of the book describes the Muddler's wedding with Sniff's mother, a briefly-seen creature named as a "Fuzzy", and Moominpappa's first meeting with Moominmamma (literally delivered up by the waves, presumably shipwrecked with her handbag intact). Nobody in Moominpappa's audience quite believes that it all really happened like he described it, until some proof arrives...
This book is great fun to read aloud, because of the shifts in narrative voice and Moominpappa's comic personality. It's probably the funniest of the series. Occasionally it is actually profound.