Always stir the paste with a plastic or metal spatula, never with the brush.
If you're making paste from scratch, use the spatula on a flat surface, like a ceramic tile or piece of glass or plastic. Squash the lump of clay and add distilled water slowly, working this into the clay bit by bit. Time spent bringing the paste to the right consistency slowly will pay off with smooth paste.
When you stir water into paste, use the spatula to mix it. Leave it to stand for a while after making it so you allow the bubbles to disperse. Sometimes, tapping the jar on a hard surface a few times will encourage the bubbles to surface.
Use good quality brushes. Always clean your brush after use and store it with the bristles pointing upwards.
Use paste sparingly on surfaces. You shouldn't have bubbles remaining on the surface of your work if you use a good brush and your paste is clean and doesn't contain foreign bodies.
If you add filings to your paste jar, this can sometimes introduce foreign bodies - like hairs and dust - to the paste. These can then get deposited on the surface of your work.
One way I've found to add a pristine surface to clay is to use syringe clay extruded onto a non-porous, clean surface, mixed with a little water and painted on.