This is one interesting request, Casper.
I will make a start to answer some of your questions. No doubt more people will join to help you out.
Food:
Flour, sugar (granulated/caster),
Salt, pepper, spices and herbs. The latter two categories would depend on what you might like to cook.
For spices, I personally like turmeric, ginger, ground cumin, cardamom, paprika, cinnamon, cloves, dried chillies, coriander seeds.
For dried herbs, I recommend parsley, oregano, basil, rosemary and thyme.
For these categories (herbs and spices), you may find that the prices in the supermarkets can be very high. If you like curries and spicy food, it is worth buying your spices in Asian shops where the prices will be much lower.
Oils: olive oil, groundnut oil (to cook Indian and Thai food)
Balsamic vinegar
Mustard (English or Dijon)
Lentils - red lentils to make dhals
Tinned tomatoes
Vegetable stock cubes
Equipment:
Try and buy stuff that is of good quality as it will last. I still have sauce pans I bought 20 years ago when I set up home.
3 sauce pans (same brand, but different sizes so that you can stack them) with lids. Occasionally, the smallest sauce pan comes without a lid. I bought a flat lid to go with mine.
2 non stick frying pans (Tefal or equivalent).
A cast-iron casserole dish (not immediately necessary but very useful) - Le Creuset stuff is good but you can get others which will last just as long but will be cheaper.
1 cast-iron roast dish (don't bother with the glass ones, they always end up tarnished!).
You don't need a food mixer but you may want to get a hand blender to make soups, smoothies. You may also need something to beat eggs.
A citrus juicer, a whisk, wooden spoons.
At least one decent big knife for general use, a potato peeler, a can opener.
Cleaning stuff:
Washing powder, conditioner.
Now, depending on whether you want to be environmentally-friendly or less so, you could just get some soda crystals, white vinegar and sodium bicarbonate and that should sort you out to clean your place.
For most people, this is not quite manageable so you may want to get a cream cleaner (such as Cif, which is very good), floor cleaner (depending on the flooring in your place), bleach...