In Dubai, if you're looking for something cultural/historical within the general vicinity of the cruise terminal, I would definitely recommend the Al Seef Dubai Creek area. It's a new retail/hotel development along the creek 10mins by taxi from the cruise terminal that has been designed to look like an old Arabian souq. I recommend it for the shopping (you can browse local crafts, clothes, jewelery etc but without the grittiness of the Gold Souq in Karama or the relentless hawking).
Al Seef is also directly adjacent to the Al Fahidi Historical Neighbourhood, home to the diwan (royal court) of the Ruler of Dubai and nearby to the Dubai Museum. It's a lovely little neighbourhood of around 50 20th century Arabian homes that have been fully restored and turned into coffee museums, perfume museums and boutique hotels and coffee shops. If you want to try out some Emirati cuisine the Arabian Tea House is also in this area and its famous amongst locals for its authenticity and quality.
In terms of tourist sight-seeing, you should definitely go for an abra ride on the Dubai Creek and visit the Dubai Frame (from which you can get great shots of the Old Dubai and New Dubai in either direction on a clear day - $15 entrance). Then there's the Burj Khalifa, but instead of paying $100 to go up to the viewing platform, just go for a $5 coffee at Atmosphere, a restaurant and bar just a few levels below the viewing deck, but with floor-to-ceiling windows.
If you want to do designer shopping, head to the Dubai Mall. It's the largest in the world by floor space and is home to the Dubai Aquarium, and of course the Dubai Fountains. If it's your thing, you could also go check out the indoor ski slope and snow park at the Mall of the Emirates ($100 entrance).
In Abu Dhabi, the Louvre Abu Dhabi is definitely recommended, as is the Grand Mosque, the largest in the country and one of the grandest in the world (the persian carpet in the main hall is the largest continuous carpet ever made). If you enjoy the idea of a desert safari, it's better to do it in Abu Dhabi as the city is surrounded by desert sands that are more untouched than those around Dubai so the drive out of the city is shorter. Abu Dhabi is also home to the Emirates Palace, one of the most luxurious hotels in the world (if that's your thing) and also the Presidential Palace (Qasr Al Watan) which from March opened its doors to the public for tours for the first time.
I live in Dubai so let me know if there's anything else I can help you with 🙂