Brushes
There are two kinds of brushes: flat (one stroke) brushes and round ones. The hairs vary from ox hair and bristles, goat, sable and squirrel to synthetic fibres. Synthetics are great for watercolour. It is springy and keeps its shape well. Genuine hair like sable is more flexible and takes more water.
The size of the brush of course depends partly on the size of the paper sheet. With practise you could probably use one brush for everything, but the different brush designs make some jobs a lot easier. Flat brushes for sharply defined geometric edges, round ones for organic shapes and calligraphic marks. A basic set could be a large and a small flat brush for washes and two or three round ones in different sizes of good quality.

Paper
You can paint watercolours on good cartridge drawing paper, but we rather choose watercolour paper. Glued watercolour blocs are great for painting outdoors, but sheets are cheaper. Cellulose paper absorps the paint less than the toothier 100% cotton (rag) paper, but it makes lifting off paint easier. Rag papers are better for working in layers and making corrections and scrubbing out things without damaging the surface. The weight in grams (gsm) is the weight of 1 square metre of that paper. In the US system the weight is in lbs, i.e. 140 lb means 500 sheets of that paper weigh 140 lbs.
Most common is 300 gsm (140 lb); 185 gsm (90 lb) is a thin paper, 640 gsm (300 lb) is almost as thick as cardboard. They come in three surface textures: HP (Hot pressed) which is very smooth; CP NOT (Cold pressed) has a moderate texture and Rough is a heavy texture. You should try different kinds of paper until you find one that suits you. Thin papers can be stretched with gummed tape, with heavy papers it is not necessary.

Palettes
All watercolour palettes are made of white ceramics or plastic. They come in handy portable folding palettes for traveling and outdoor painting, and larger ones for use in the studio. Some have lids so paint doesn’t dry out. Most important are one or more large mixing areas for your diluted paints. Which ever type you choose, fresh tube paint is the best to work with, although dried paint is still re-soluble for use.