Here's some advice on techniques and a few tips on how to varnish a piece of furniture.

Choose a flat, very thick, fine hair paintbrush. A cloth can be used for really fragile items. You will need fine-grained sandpaper for sandpapering. If you want to imitate the professionals, you're best to use steel wool (the finest grade of steel wool is 0000-000).

You should never apply varnish in thick coats. Start with a thin coat of varnish, applied in the direction of the grain. Wait for it to dry, sandpaper it very gently using a very fined-grained sandpaper (400 grit emery paper), always in the direction of the grain. Get rid of the dust, then repeat the whole process twice.  Three very thin coats of varnish will make for a very resistant varnish finish and a piece of furniture that will look so much better.

In the art of cabinet-making, varnishing is done in a damp environment. It is vital to work in completely dust-free conditions  (so that there's no dust in the air to stick to the varnish that's been applied). Don't hesitate to wet the floor around you before you start to apply the varnish, this will ensure the dust sticks to the floor...

Matte or gloss? It's really a matter of preference.
However, for the sake of your health, and that of the environment, it's best to use an organic varnish (made from natural products) which can be bought in organic outlets or online.
At the very least, buy a varnish that's green seal-certifed meaning that its VOC (volatile organic compound) content will be very low (compare with other big name brands of varnish that are put to the fore in hardware stores, you'll be surprised!).

Be careful, varnishes produced by brands specializing in arts and craft products may be just as harmful. Avoid solvent-based varnishes (that are washed off using white spirit rather than water) such as the old kind of Pébéo varnish, which was an attractive amber color, but which gave off a very strong, unpleasant odor even long after the object varnished had dried.
You should always apply varnish in a well-ventilated room, away from children and pregnant women. Wait for a few days before bringing the varnished object or piece of furniture into the house.