A good knife
No kitchen should be without a good chef's knife. I cringe when I see people using serrated chef's knives or thin carving knives as their main chopping knife. Honestly I don't think that is safe, despite claims made by the maker. Too much force needs to be used, increasing the chance of the knife slipping.
What every kitchen needs is a 10-inch chef knife. There are many brands out there, so it can get a bit confusing. I use Sanelli, which I got in culinary school, but Henkle (with the two man symbol, not one man), Victorinox and Wustof all make great knives. Locally you could try Grohmann knives out of Pictou, though I have never used one personally so I'm not sure about the quality.
A 12 inch frying pan
Every kitchen needs one of these, and a heavy bottom is a must. The size may seem large for just one or two people, but you never have to worry about crowding food on your pan. The heavy bottom negates any hot spots a burner may have, and makes sure heat gets to the food evenly.
Also skip the non-stick. A regular frying pan can go from the stove to the oven easily, which most non-stick can't. Regular frying pans brown food better, and create a rich brown "fond" on the bottom. With a little water and spices, that fond becomes a wonderful pan sauce to add to your meal.
Buy a small non-stick for eggs though, or else you will never have easy over again.
A good bottle of olive oil
The way some people talk about extra virgin olive oil makes it seem like something for culinary masters, however I think that's a mistake. Buying good olive oil is just like buying the right cut of meat. You wouldn't try to make a stew out of prime rib, or grill a piece of eye of round.
With olive oil, it is best used for drizzling on foods, marinades and light cooking duties. It just seems to perk up the flavour of things. It can be used for frying, though you lose a lot of the flavour with the higher heats. Canola Oil is what I keep nearby for things that need that high heat.
As for buying it, you need to do a bit of hunting. The stuff from the grocery stores just doesn't cut it, sorry.
Olive oil is a little like wine, in that you can have a preference for a certain brand. My favouite brand is Korinthia, which can be found locally at Snug Harbour in Lanes Privateer Inn. I hear there is a place in Halifax called Liquid Gold that specializes in olive oils, which I will some day go to in order to broaden my horizons a bit.
A big cutting board
Mine is 12 by 18 inches, and I adore it. The work surface is big enough for any chopping task, and it stays put when I want it to. Best of all, they're cheap at around $10 to $15. Brand doesn't matter too much, just be sure it doesn't seem slippery to the touch or gouges when you press your fingernail into it. Slippery is dangerous in the kitchen, and if it gouges too easily it won't last very long.
A subscription to Cook's Illustrated
I feel like I keep shilling this magazine, but it really is one of the best. So many other magazine recipes are disappointing, but even the bad ones from Cook's Illustrated at least can be traced back to me just not liking the flavours.
I think this magazine, which can also be purchased as an online subscription, is doubly useful to the new cook because it explains why they did things the way they did. Teaching someone to cook is more valuable than just giving them a recipe.
A water filter
OK, so this doesn't quite fit in, but I wouldn't leave this out of any kitchen of mine. Anything you need water for in the kitchen- pasta, bread, etc. - should be filtered. Though this is important in municipal supplies to get rid of the chlorine, even well water needs filtering I think. It eliminates any off tastes the water may have, which can come through when cooking something.