They are clearly all from the same lexical field. What you need to know is what restrictions are placed on their use.
Equipment usually describes (all) the necessary articles for a purpose. Thus you would say:
'A lot of equipment was needed for this mountaineering (or camping) expedition.'
Apparatus is similar in meaning to equipment, although it tends to relate to particular contexts. Thus gymnastic apparatus refers to all the apparatus you would expect to find in a gym. It tends to be used in scientific or technical contexts:
'The chemistry lab was full of the apparatus needed for a range of experiments'.
For tools and utensils it is also very important to know which other words they collocate with. For example, we talk about garden tools for a gardener and work tools for a carpenter, but kitchen utensils for a housewife or house-husband.