Knives: A good Chef's knife, paring knife, and serrated (bread) knife.  Which brand is so hard to say, depending on your budget and preferences.  A whole article can be written about this.  But, here is a quick list of some faves:  

Sauce Pans:  Small (2 qt.) and medium-sized (4 qt.).  A heavyweight is usually a sign of multiple layers of metals and quality.  Copper is the best conductor of heat, but needs to be polished and is very expensive.  It’s the best you can buy.  Aluminum is next best but reacts with food, so it needs to be anodized or covered with another metal.  Great pans will have a core of copper or aluminum on the bottom AND sides.  These metals will be sandwiched between stainless steel ( a poor conductor of heat on its own).  Surprisingly, Ikea has some great pans with these qualities.

Large pots with Handles:  A medium (5 or 6 qt.)  and large (10 to 12 qt.) Stock Pot.  Also, you need a 10 to 12 qt. Dutch Oven, which can double as the larger stock pot.  The Dutch Oven should be either Cast Iron or Ceramic coated cast iron like Le Creuset or a knockoff.  Oval or round, it doesn’t matter.

Pressure Cooker:  Get at least a 6 qt.  This can also double as your medium stock pan.  I prefer the mechanical ones like Kuhn Rikon to electric because the latter seems to take longer to cook.

Sauté/Skillet Pans:  A 10-inch and a 12-inch cast iron skillet are amazing pans.  Again, whole articles can be written about these.  They are a great value and can last forever.  Besides these skillets, I like keeping at least one 8-inch Non-Stick Skillet and Crepe pan too.

Tongs:  A large one and an extra large one for grilling are good to have.  Chopsticks work too!

Mortar and Pestle and/or a Spice Grinder

Cutting Boards: one small and one large wooden reversible cutting board with channels for drippings.  You should also have one or two plastic cutting boards for raw meats.  Also, in small and large sizes.  You can put the plastic ones in the dishwasher.

Grill/Griddle Pan:  I prefer a cast iron one, but it’s not necessary.  Often these are reversible with a flat side griddle in addition to the grill side, which comes in handy for making pancakes, etc.

Universal Lid:  By far the most asked about on this list.  These lids fit more than one size pan, which is great for not only saucepans, but also my sauté/skillet pans.

Baking/Roasting Dishes:  Preferably white porcelain. Also, aluminum pans are good too.  At least one 9-inch x 11-inch.  I also like having a small roasting pan (8 in. x 10 in.) and a really large one (13 in. x 16 in.)

Pie/Cake pans:  Two 9-inch round cake pans and also a 10-inch springform pan.  Optional items: one 9 in. x 5in. Loaf pan, one 9-in. pie pan, one 10-in. tart pan, and one muffin pan.

Sheet pans:  At least one half sheet and one quarter sheet.  Two of each is best.  Lids are great.

Food Processors:   A mini 4-quart Cuisinart is so useful.  Also, a 14 cup processor is good to have.  You can make pasta, desserts, etc. in these.

Salad Spinner: Wet greens don’t allow the dressing to stick

Whisks: Large and small

Flat Wooden turners (spoons or spatula):  these are great for scraping stuck on bits in your cast iron pans while cooking.

Spatulas & Dough Scrapers:  Silicone for baking and scraping bowls and pan sides.  Fish spatulas for turning foods.

Skimmers:  A fine mesh one and a wire, spider skimmer are a must.

Microplane Grater

Funnels:  Small and Large

Weight Scale

Dry Measuring Cups and spoons

Liquid Measuring Cups

Vegetable Peeler

French Rolling Pin (no handles)

Meat Thermometer:  Digital ones like pen style instant-read are great.  It needs to have a tapered edge.

Hand Mixers (Stand Mixers are worth investing in if you become a baking fiend)

Small and large Colanders and strainers:

Small, Medium and Large Stainless Steel Mixing Bowls

Ramekins:  4 oz. and 10 oz. sizes are great to have.

A good Corkscrew (double hinged)

Can Opener

Butter/Basting Brush (get silicone so the bristles don't come off in your food)