Here is a list of the most valuable equipment to you in the kitchen as a home cook. I will be doing an electrical/gadget blog post soon but this will cover your basics!
It really is worth spending the money on good quality equipment, it'll last longer and ultimately work more effectively.
It's better to have a smaller selection of high quality items than cupboards stuffed with cheaper pieces which frequently need replacing.
Knives (a large chef's knife, a small paring knife and a large serrated knife)
A large non stick frying pan
A set of heavy bottomed saucepans (set of 3, small, medium and large with lids)
A decent wooden chopping board
Weighing scales, measuring spoons and measuring cups
At least 2 oven sheets
A good vegetable peeler (my favourite is the OXO speed peeler)
Nest of mixing bowls
A couple of wooden spoons and a rubber spatula
Set of round metal pastry cutters