There are affordable hotels around Barsha area - Holiday Inn, Novotel, Ibis Hotel, Citymax Hotel. They all cost less than AED 500 / USD 130 a night and have pools. The hotels near the beach are mostly (if not all) five stars. The ones in Barsha are not too far off, just a 5-10 minute taxi ride from Burj Al Arab, which is beside the waterpark and the public beches. Taxis are cheap anyway, it will cost around AED 10-15 (USD 3-5), plus it's near the metro station so it's pretty easy to get anywhere. It's also beside the Mall of the Emirates.
To be honest, there's nothing really much to see in Abu Dhabi, although the Grand Mosque is beautiful, but that's pretty much it - unless you want to go to Yas Waterworld, which I heard is pretty amazing. If you really want to do it, I suggest just one day in Abu Dhabi, go to Grand Mosque in the morning and the waterpark in the afternoon, then back to Dubai by evening.
In Dubai, you can pretty much find all the must-see by searching Google (desert safari, dhow cruise, Burj Khalifa view from the top, etc). Although one thing I highly recommend that isn't often mentioned online is the Bastakiya cultural breakfast or lunch. Check www.cultures.ae - it is really worth it where they serve Emirati food while you're sitting in a circle and there's an Emirati moderator who directs a Q&A portion where you can ask anything about Islam and Emirati tradition. Definitely worth it!
Additional notes:
- summer has arrived pretty early, it's around 37C so be prepared
- if you're doing the view from the top at Burj Khalifa, book it online as it's much cheaper when you book in advance, around AED 110
Enjoy Dubai! :-)