You can do up to 2560 by 1600 on the Air. Some newer monitors have a display port option so you just need the cable, if it doesn't come with one. You can also use a HDMI adapter as most screens have HDMI, and then you can reuse it for TV use.
A monitor with a IPS LCD panel gives you better colours and viewing angles (Samsung calls this PLS). Some have usb hubs, sound, rotation, adjustable height, etc. It really depends on what you need.