Is there any other kingdom than the Mushroom Kingdom?
According to the Super Mario Wiki there are quite a few!
If we stick to just main series Mario games, you have:
the Beanbean Kingdom from Mario and Luigi SuperStar Saga.
Birabuto Kingdom, Chai Kingdom, Easton Kingdom, Muda Kingdom from Super Mario Land for the Game Boy.
Sammer's Kingdom from Super Paper Mario.
Flora Kingdom mentioned but not actually see in Super Paper Mario.
The Dry Dry Kingdom mentioned in Paper Mario, but not actually seen.
Sprixie Kingdom from Super Mario 3D World and Toad's Treasure Tracker.
Nimbus Land from Super Mario RPG, which although it is called a Land and not a Kingdom, does have a King and Queen.
There are also various Lands, Towns, Towers, Forests and other areas not listed here, as you were only asking about Kingdoms specifically. Donkey Kong Jungle Beat has various fruit themed Kingdoms as well. There is also the Baseball Kingdom mentioned in Mario Super Sluggers.
Also as pointed out in comments, if we believe Luigi when he mentions it in Thousand Year Door, there is also a Waffle Kingdom.
Yup, there are.
Super Mario Land takes place in Sarasaland, and is the country Daisy rules over.
Super Mario Land 2 takes place in Mario Land, although I'm not sure if it's located within the Mushroom Kingdoom or if its an independent country.
Super Mario Galaxy 1 and 2 introduces many new planets, and by extension many new kingdoms/land/countries/bakeries/et cetera as well. Unless the Mushroom Kingdom recently became a galactic empire, I think it's safe to say all of these planets are located outside the Mushroom Kingdom.
Mario is Missing takes place..well, all over the planet earth. The first Mario Golf game takes place on the planet Earth as well. There's also the Donkey Kong games, which was likely intended to take place in Earth as well (I don't think the Mushroom Kingdom was even conceived until Super Mario Brothers).
Paper Mario introduces the land of Star Haven, and Wario Land: Shake it! introduces the Shake Dimension.
I'm sure there are more I've missed, but as you can see, there's a lot more to the Mario Universe than just the Mushroom Kingdom.
And we can't forget Nimbus Land, the misleadingly named cloud kingdom from Super Mario RPG, Legend of the Seven Stars, ruled by King and Queen Nimbus and
usurped by the evil and vain Valentina.