Dinoflagellates are relatively easy to maintain at home, requiring as little care as a houseplant, except that these "plants" produce bright blue light when shaken at night.
For more technical culturing information, we have another page about Growing Dinos in the Lab
Also see our page documenting a red tide of dinoflagellates.
Where to get bioluminescent dinoflagellates
[Disclaimer: we have no ties to these companies and list them here as a service, not an endorsement.]
A very bright and very tolerant species is available via mail through this source: (Note: this species is non-toxic).
Sunnyside SeaFarms sells Pyrocystis fusiformis