Programs: UA 1K, QR Pt, EK Au, TK E+, SPG LifePt, IC Spire/RA, Hyatt Dia, Shang Dia, Marr Pt
There is almost nothing to do on Yas Island. Besides Ferrari World (I am not sure about the public beach), there are no attractions.
You are about half an hour away from downtown Abu Dhabi and 75 minutes away from Dubai -- and transportation is easy by taxi (except to spend about AED 300, or $80) or rental car -- so, from that standpoint, Yas Island is not a bad base for a week-long trip to Dubai/Abu Dhabi.
But don't expect to spend the whole week on the island. If I were you (as it seems like you haven't been to Abu Dhabi or Dubai before), I might spend two or three nights on Yas Island, to visit the island itself as well as Abu Dhabi, then spend the remaining nights in Dubai.
Yas is 15 minutes from AUH, 45 minutes from the Marina end of Dubai and probably a good hour to DXB. I agree with mecabq: a week is a long time to spend on a desert island with one golf course, one tourism attraction and 5 hotels. I'm not sure when the new Yas beach will be open - but the Corniche beach is good on the main Abu Dhabi island (a bit of a trek from Yas though).