Sand the surface as smooth as you can and take out all stains or blemishes prior to starting. Dust thoroughly. The fine grit left when sanding is a more common cause of rough-looking, “sandy” finishes.
A good quality bristle brush, absolutely clean, is a must for great varnishing. The brush must be soft and pliant with plenty of flagged bristles. Do not dip it into the varnish more than a third of its length, and always flow the varnish on generously with long strokes put on parallel to the grain. Most crucial of all, do not wipe excess varnish off the brush by dragging the bristles across the rim of the can. Rather, tap the tips of the bristles softly against the inside of the container above the surface of the liquid. Wiping across the rim induces tiny bubbles to form in the fluid and these would make it nearly impossible to achieve a smooth finish.
Air bubbles on the surface likewise can be caused by pressing down excessively hard on the brush. Varnish must be brushed on with just light to moderate pressure. After covering the panel in one direction, make a cross-stroke right away by brushing across the grain in right angles to the original strokes. Finish up by lightly stroking a 3rd time with the grain, using an virtually dry brush and touching the surface with the tips only. This crisscrossing method will almost eliminate all brush strokes so that a uniform application is assured.
Varnishing must never be done in a dusty location, or in garages or basements which are moist or cold. If possible, lay pieces horizontally and always take out all hardware and knobs prior to starting. Chests or cabinets that bear drawers are best done with the drawers taken out so that each one can be placed upright with its front in a horizontal position.
If more than one coat of varnish should be applied, the first coat must be thinned with about ten per cent turpentine. Consecutive coats may then be applied as they come from the can. For a super-smooth finish, sand gently between coats. Use #2/0 very fine sandpaper and rub along the grain. Remember to dust thoroughly prior to applying the next coat. The best way to grab up the dust is to use a tack rag, an especially treated sticky cloth which is available at several paint and hardware stores.
Paint and Varnish Facts and Formulae by John Norwood Hoff
Old House Journal: Restoration Techniques