Which room is the play room? The study?

Sorry! I forgot it was labeled as the study. Yes it is the playroom

How old are the kids? What activities will they do in this room? What things/toys will they store there?

If the desktop will be used by the kids you want the monitor to face the door, so you can always see what they are surfing with a quick glance into the room.

Do you want true built-ins, or would you prefer this room remain flexible? I ask because, we found for our family, we really liked the Expedit line from Ikea for around our family room/play room. The cubbies are a good size (they hold Lego containers really well); they are inexpensive, and easily changed when life stages change... You can choose the single "height" style when on its side and put a cushion on top for a "bench seat"/window seat. You can get the 2x2 or 2x4s or 4x4s or 5x5s. For us, their modular design worked really well... But, they aren't built in (though people build them in by adding crown, etc).

Given the way technology changes, I wouldn't want a true built-in designed to hold a computer.

If we were to add built-ins to our playroom, I would do a variation on IKEA's Trofast. We built a cheap DIY Trofast unit and the organization is fantastic and flexible. The Trofast bins come in several sizes, slide in and out like drawers (but can easily be pulled all the way out) and have matching $1 lids.

Thank you guys so much for the ideas! I showed your ideas to my husband and he agrees that ikea is the way to go. I have never bought anything from ikea. We live about 4 hours from a store. Do you recommend just ordering online or going to the actual store?

I'd go to the store. Can you even order online?
I'd also check before you go to make sure they have what you want in stock... Or, do you have a friend who is local to one who can do the shopping for you? (where are you?)