I have a small camera and this is the first time I'm filming anything that is serious. I'm wonder if there is a cheap way I can smoothly shoot footage in a room while moving without a camera dolly.
You want something like a Steadicam.
The idea is to add weight to the camera assembly so that it is stabilized by an increase of moment of inertia. For larger rigs, the entire thing might be supported by the cameraperson's body on a harness.
For smaller applications, the mount point of the camera is right above where you grip the assembly, and the pivot point is at the top of this grip. Then, a rigid strut extends from the mount point to below the grip, and weight is added at the bottom of this strut so that the center of gravity of the entire assembly (camera included) is at the pivot point.
A steadicam is one solution, but there are others. You could use a slider dolly, camera crane, jib, shoulder mount, or fluid head tripod, depending on the movement you want.