I have the ATT LTE version of the Note and it is very fast for internet browsing. My old 3G Samsung Galaxy S Captivate was like dial-up in comparison. The screen is big enough for most browsing activity. I have the builtin browser set to be recognized as a mobile device and I use Opera set as a desktop device for certain sites where appearing to be a desktop device has some benefit.
The picture quality for video is amazing. Before getting the Note I had an old 7" tablet I used at the gym while doing my seventy minutes on the treadmill four times a week. The Note is a little smaller, but the picture quality is so much better it more than compensates.
For my usage, the battery life is more than acceptable. 70 minutes of video at the gym, 70 minutes listening to audiobooks in the car during my commute, 30 minutes using the Kindle app to read books, and maybe fifteen minutes of checking email and other misc surfing and by the time I go to bed, it will typically have about 65% of the battery left.