With the serrated leaf edges like that, your tree is most likely Chinese elm.
Chinese elm is a sort of in-between tree. Temperate trees (evergreens like juniper and deciduous like maple, oak) have to be outdoors all the time. Tropical evergreens like ficus make good indoor trees. Chinese elm is a subtropical. Some people do keep them indoors. If yours is used to being indoors, I wouldn't put it out now, if it is already cold and snowy where you are. To winter over outside, they need to go through the autumn, to gradually toughen themselves for winter. After the weather warms up next spring, you can gradually transition it to outdoors.
Chinese Elm trees are very flexible about their environment. If adapted
properly, they can be grown indoors year-round. If hardened off to
the cold gradually, they can drop their leaves in the fall and be
considered deciduous trees, so you have a few options. However, it is
important to find out how your Elm has been growing recently. Some
Elms come from southern China, and have never experienced cold
conditions. It would be dangerous to keep these Elms too cold the
first year or two. Other Elms have gone to freezing and dropped their
leaves for many years; as a result, these Elms may not be too happy
indoors for the first year or two. If you can not determine where your
Elm has been, then keep it outside for the summer and bring it inside
for the winter. You can bring it in when the temperatures are around
50Â°F. If kept indoors for the winter, a cooler location is preferable
(50Â°Fâ€“65Â°F). Chinese Elms may drop some leaves in the winter due
to the natural decrease in light. This is normal.
So I don't know if those branches are dead. Break off a twig end and see if it is flexible and green inside. If yes, then it just dropped leaves, but the branch isn't dead. If a couple of the branches are dry and brown all the way through then just make a little scratch on the trunk bark with a fingernail. If it is green under the bark, then the tree is still alive even though some branches died.
At this point, since it has been dropping leaves, I would let it go on into dormancy, which is what it is trying to do. Find the coolest spot in your house (my old house had a coat closet next to the outside front door which didn't get much house heat, didn't freeze but stayed very chilly). Put your tree there. Be sure it gets some light as long as it has leaves (if it's a closet with no light, you might need to put a little lamp on it) but don't worry about it having lots of light. If it drops all the leaves, you don't even have to worry about light. Water it infrequently, just enough so it doesn't dry out all the way through. Do NOT fertilize. Let it rest until spring.
In the spring you will need to repot in to good bonsai soil. I don't know why it seems that bonsai trees are always sold in soil that is very bad for them. Here's a lot of information about bonsai soil with links to more:
https://www.helpfulgardener.com/forum/vi ... =36&t=3422
I would summarize the soil info by saying that bonsai soil would be mostly pine bark and mineral grit (fired clay, granite grit, ground volcanic rock, etc), with maybe a little bit of humus. It is supposed to be very loose and free draining, not hold too much water.
Never water bonsai on a schedule. Water when they need water, which will vary with temperature, humidity, season, how fast the tree is growing, etc. As noted when the tree is dormant and not growing, it will be watered infrequently.
Here's some good information about bonsai care including watering and how to know when to water: https://www.helpfulgardener.com/forum/vi ... =36&t=1479
Best Wishes for your new Chinese Elm bonsai. Welcome to the Forum!