What is a good starter bonsai tree? Are there any good beginner bonsai that
can stay indoors year round?

It is spelled bonsai, singular & plural.
The best type of tree depends a lot on where you live. In the Northeast US,
species of tropical Ficus are usually recommended for beginners. You can buy
them in any garden center. However, you should plan on putting it outdoors for
the summer, even if only on a balcony.

Iris,
Central NY, Zone 5a, Sunset Zone 40
"If we see light at the end of the tunnel, It's the light of the oncoming
train."
Robert Lowell (1917-1977)