Pots and Pans:

A minimum set includes a small frying pan,a one quart saucepan with cover, a two quart saucepan with cover, a large frying pan with cover, and a large pot (for spaghetti, stew, pot roast, or homemade soup.) If you are cooking for more than one person, get toe each of the frying pans and saucepans. Covers are optional for the frying pans, but try to get a cover to fit at least one.

Good quality pans spread the heat evenly so that food doesn't burn in one spot while not heating well in another. The bottom of a good pan will sit flat on the stove. Cheap pans wear out fast and end up costing more in the long run. Stainless steel with a porcelain coating or a copper bottom are excellent. Stay away from aluminum pans. they get bent out of shape easily and don't last. There have also been some health warnings about cooking in aluminum, but no real proof of any danger.

When using pans with a non-stick coating, don't turn the heat up any more than necessary, and only use wooden or plastic tools. Wipe clean in soapy water. A plastic scrubber or nylon net can be used, but never use steel wool pads. If you don't burn the food onto the pan, and wash it while the food is still fresh (don't let it sit around and dry out), it will be easy to clean. If you have birds as pets, you may want to avoid non-stick coatings. If you use them, be sure not to overheat them. The coating can release fumes that can kill a bird.

Some other useful tools are a slow cooker, a wok (for Chinese cooking), toaster, microwave, blender, food processor, electric mixer, and an electric knife. There are also bread machines available at reasonable prices that produce a fresh hot loaf of bread in an hour or two.

Basic Gadgets:

You will need some wooden spoons. Get a selection of sizes and handle lengths. These are always handy and they won't damage your non-stick coatings. Unlike metal tools, they won't get too hot to handle.
You'll also want a collection of spatulas in different sizes and shapes. The lightweight spatula that will flip an egg may not be strong enough for a hamburger. Sometimes a narrow spatula is all that will fit in a pan, and sometimes a wide one is needed to get under whatever needs to be turned or lifted.

Whisks come in various sizes. They are very handy for mixing lightweight ingredients quickly, and are easier to get out and later to clean than a blender or mixer.

Get a decent set of knives. Cheap ones make the job of cutting more difficult and they wear out fast. With the knives, you'll need a good cutting board. A cutting board provides a flat working surface and protects your counters. Plastic cutting boards are easier to clean than wooden ones. Never cut vegetables that will be eaten raw on a cutting board that was used to cut raw meat unless you thoroughly clean the board first.

You will also need a selection of measuring tools. Measuring cups for dry ingredients and measuring spoons come in sets. Get a set of each. You will need a measuring cup for liquids, too. A one-cup measure is probably sufficient, but a two-cup and a quart measure are also useful. If you want to be precise in your measurements for dietary requirements, buy a food scale and measure your portions until you get used to the sizes and can estimate accurately without weighing everything.

A lot of canned products are now being packaged with pop tops, but get a good can opener for the ones that aren't. There are some electrics that sell for reasonable prices and do a good job, but get a manual one too. When the power goes out, it's nice to still be able to open those cans.

Spices and Condiments:

Salt and pepper go without saying. Some dried minced onion peps up a lot of recipes. Garlic is available fresh or in dried, chopped form or as a powder. Caution: make sure you know the difference between garlic powder and garlic salt. Garlic powder makes great garlic bread; garlic salt makes VERY salty garlic bread.
Oregano or "Italian Seasoning", gives an Italian flavor to anything. A bay leaf cooked in stew brings out the flavor of the other ingredients. Paprika is a red powder used mainly for appearance, but it has been linked to some health benefits. If you like things hot, try sprinkling some cayenne pepper or chili powder on your food.
Basic condiments include ketchup, mustard,relish and mayonnaise. There are all sorts of mustard varieties available. You can probably find traditional yellow mustard, Dijon, spicy brown, honey, and hot Chinese mustard, just to name a few. Explore your local market and see what looks good. Be adventurous, but buy the smallest size in case you don't like it.
Other condiments to try are steak sauces, barbecue sauces, tartar sauce, and various stir-fry sauces. Read the labels. If sugar is one of the first few ingredients, forget it. Ingredient labels list the ingredients in descending order. The main ingredient is first, then the others, with the least one last.