There are a variety of eyeshadow tools you can choose to use. These range from sponge tip applicators, brushes made from real hair ie. sable, synthetic brushes, cotton buds/q tips and your fingers.
When applying cream bases and cream colours I usually opt for a synthetic brush as it will be easier to clean product from than a real hair brush or I will just simply use my fingers.
For powder shadows I tend to prefer a real hair good quality brush. There are a wide variety of brushes to choose from. Generally a smaller brush will be used to create more definition and achieve a more precise application. A larger brush will achieve a softer, lighter payoff of colour.
A firm brush adds stronger colour to the eye, a soft brush therefore adds a light amount of colour. Soft, fluffy and tapered shaped brushes are great for blending colours. Very small tapered, rounded brushes are great for applying colour around the lash line.
Personally I tend not to use sponge tip applicators but they can be used to apply colour to the lid and are useful to pack colour on in layers to achieve a stronger colour. Cotton buds are excellent for cleaning up the edges of your eyeshadow especially the pointed cosmetic bud variety.
I would reccommend having a least two eyeshadow brushes. A firmer one for applying colour and a softer, slightly biggger brush for blending colour.
Small and firm synthetic pointed or angled brushes are best to apply any form of eyeliner.
I highly reccommend looking at the following brands when searching for good quality brushes:
MAC, Bobbi Brown, Real Techniques and Daniel Sandler.