One of the easiest mosses to grow indoors is Java Moss.
This is an aquatic moss that can grow well on a very wet substrate as well as in water. It is tropical and can handle the heat of a home. It is one moss that you would not have to worry about over-watering. It can do well with 200 or less foot-candles of light.
However, I think you are thinking about terrestrial mosses.
If you have the means to do so and can provide the necessary conditions just about any commonly encountered terrestrial moss can do well inside in a terrariusm setting. The secret is in knowing how to grow it and to provide the right combination of light, moisture, and coolness. To learn you really need to learn how the moss you may be interested in is growng in nature. Take a digital and take images of the kind of moss that interests you. Take images at least once a month. Make notes on light quality (foot candles if you have a meter), humidity or wetness, lay of the land, soil characteristics and changes to the moss covering (such as leaf fall). This is really a matter of learning what is happening. Take a thermometer and measure the air temps.

I do not know if there really is a fastest growing moss.
If you have the conditions right then most will probably grow as fast as any other. I think it depends on how much you have and how you set up the planting (as small plugs or as a sheet or ???).

I would recommend trying sphagnum. It is a very flexible moss and while it grows in cool spots it also grows in the tropical areas such as Florida. I am growing some in my greenhouse that came from New York on a shelf that is just under the roof and it gets the most light it can. You can buy live sphagnum over the net and you can easily propagate it on a peat moss substrate.

Whatever you do do please keep notes or images as a journal on the moss as I would like to read about your efforts.

Selaginella is moss like. If you are thinking Japanese garden/Bonsai lush mat moss then you have to provide constant moisture. Very porous rock like old limestone or lava set in a shallow(2-3") ceramic tray of water will turn green with moss-as long as the tray is never allowed to go dry. To push it faster seed with groundup moss of your choice.You might find ferns like maidenhairs sprouting after a few months.
I did.