What should I look for in a new kitchen knife set?

Typically, you don't want to buy a 'set' of knives. You're better off buying the knives that you specifically want. Often, you'll want to get a mix of knives from different manufacturers, so that you can get the best knife for each purpose.

As for what to look for ... I'd have to say that the main issues these days are weight, handle, and how the knife is made:

• Most knives should balance near the front of the handle. If it doesn't, you'll end up doing more work at the wrist to keep the knife in control, which will tire you out. (exceptions are small paring knives, cleavers, and maybe exceptionally long slicers).
• The blade should be full tang. This means that the blade continues all the way to the end of the handle, giving it a better connection to the handle while improving the balance.
• The handle should be comfortable in your hand. This will of course vary per person, as we all have different hand sizes.
• The handle shouldn't be too slippery. If it is, you increase the chance of cutting yourself.
• You should avoid ceramic blades unless you're specifically going to be cooking a lot of food that will discolor when cut by steel.

Now, there are some issues that are personal choice:

• Most home chefs should avoid carbon steel blades. They have their advantages, but they need extra maintenance or they'll rust. If you didn't grow up using them and are used to immediately cleaning & possibly oiling your knives and can sharpen your own or have a reasonably priced sharpening service nearby, I'd recommend steering clear of them.
• Many people prefer forged vs. stamped blades. Although the sign of a forged blade used to be the bolster (the thick bit of metal just before the handle), newer manufacturing techniques can add a bolster to a stamped blade. Stamped blades are thinner and more flexible, which is an advantage in boning knives. They also tend to be considerably cheaper, which is a huge advantage when you're starting out. Not all stamped blades are bad; I believe that the the Victorinox Fibox chef's knife has won America's Test Kitchen's round of tests. On the high end of forged blades are folded blades, which can be quite expensive but hold a great edge.
• Knives for slicing should be fairly heavy. They shouldn't be cleaver-heavy, but you want the weight of the knife to do the work, and not have to be pressing forward with your wrist as you slice. But if you're going to be torting cakes, or other cases where you're slicing horizontally, you want the slicer to be much lighter.
• The angle of the blade affects how sharp it is, but also how often it needs to be sharpened (aka. 'edge retention'). A very small angle creates a very, very sharp blade, but if you nick a bone or something hard, it's more likely to be damaged. Even just leaving the knives in the sink is risky with some exceptionally sharp knives (the weight of the knife itself can break off the tip). Japanese knives tend to be a smaller angle than European blades, but some of the European manufacturers have started putting out sharper knives. If you have the budget, you may want a variety of angles (a small angle for vegetables, a wider angle for meats)

These are all general characteritics. It's take much longer to go into details about each type of knife. See for instance What should I look for in a good, multi-purpose chef's knife? . Serrated knives are getting trickier, as there are now different types of serrations (regular, micro, wavy, etc.); even micro serrated blades (eg, Ginsu) have their advantages.

Update: I forgot to mention the personal choice 'granton edge' (dimples) or holes along the side of the knife near the cutting surface. It can reduce the amount of drag that you get when going through solid but moist foods (apples, firm cheeses, etc.) by breaking the adhesion between the food and the blade.

Also, when it comes time to buy knives, consider asking at the store if they'll allow you to test the knives; bring some carrots and apples just to get a feel for how the different knives feel in your hand when actually cutting. (if they won't allow you to actually cut something, they might still let you try chopping and other tasks on an empty cutting board.

Don't buy a set. You've had a set for a while, so you know which knives you gravitate to. Spend your money there. Don't waste your money on knives you don't use 'cause they look nice in the block.

America's Test Kitchen did a show on knife sets recently, and reported similar findings to what has already been noted here by the other commenters. I am only including this as it is technically a set, but you simply buy the components based on their own merit and not what the manufacturer is trying to get rid of/sell you.

Per America's Test Kitchen:

• Wusthof Classic 3.5" Pairing Knife: $40
• Victorinox 8" Chef's Knife: $30
• Wusthof 10" Bread Knife: $120
• Victorinox 12" Slicing Knife: $58
• Victorinox 6" Boning Knife: $20
• Shun Kitchen Shears: $70
• Bodum Bistro Block: $50

http://www.amazon.com/Cooks-Illustrated-Kitchen-Carte-Knife/lm/R27FG2C1FMTA16