Internet access? - Yes, but it is limited to the ship's intranet linked computers. And there's like say, 2 per a division of 30 personnel.
Can you bring your laptop/tablet? - Yes
Can you bring your laptop/tablet to do course work on? - If you need access to the internet, no. You are not allowed to directly connect any personnel electronic devices to the ship's network. Also, when I retired in 2011, the carriers didn't have any wireless network onboard.
Me personally, I did my initial sea tour without doing any college courses. After I transferred to shore duty, I got my associates. 2nd sea tour, no classes. 2nd shore tour, finished a bachelor's.
There are courses offered on carriers. The majority of them were on-line, with several computers (like 3) set aside in the ship's library specifically for these courses. However, there were 2700 personnel in ship's company, with another 1300 in the airwing on deployment. Add in a watch rotation of 5 and dimes (5 hours on watch, 10 hours off) or 5 and 15s (15 hours off), during which your "off" time really isn't off since you need to sleep, attend training, do maintenance, work on your quals, do admin work, etc. And yes, I realize my surface watch rotation was probably way better than my submarine brethren

Just some tidbits about life on a carrier underway. Best of luck to you and thank you for volunteering to serve your country!