What features should I look for in a new wheelset? I am building a custom bike and I can spend $500-$1200 on wheels.  What are the best features on a set of road wheels?

It would be nice if they were lightweight for climbing but I live on the flats so aerodynamics are also important.  Basically an all around road wheel.
Current rim depths tend to be between 20mm and 80mm.  You can choose along this spectrum to trade weight for aero, but for general use 30mm or 40mm is probably reasonable.

Hand-built wheels are an advantage if done well, because they are less likely to have broken spokes.  There are plenty of custom wheel builders out there, plus some mass-market manufacturers who advertise hand-built wheels (obviously they'll automate what they can, but hopefully the important parts are done by hand).

For hubs you will find two types: cartridge bearing and cone bearing.  The former are adjustment-free and you generally can replace the bearings if needed, whilst the latter require adjustment to get the perfect spin.

You'll probably want butted spokes of some kind.  I dislike aero spokes (with flattened midsections) because they are painful to work on (you inevitably whack your hand with the spokes when working on the bike, and bladed spokes can actually cut you).

A recent trend is toward wider rims, which allow more tire volume and are sensible if you will run 25-28mm tires.  It's also easier to install and remove wheels if the tire is not much wider than the rim.

In your price range you should be able to find 30mm deep wheels around 1500 grams, give or take.