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The sea shore flow (link below) breaks down sea and shore time for all ratings. Keep in mind this is the "standard" flow time for an balanced career. See wikipedia for a good breakdown of all ratings
The most sea intensive (not all inclusive) would be your aviation boatswain mates, boatswain mates, engineer ratings, and combat systems ratings (see the sea shore flow). Now, that being said, if you are looking to spend 20 years afloat, the Navy definitely has no problem with that. Like you said, there are many people trying to avoid sea duty but few trying to maximize their sea time. There's also "sea duty incentive pay" (see below) that gives bonuses for ratings choosing to extend their sea tours. I know an EM1 who spent 12 years at sea, and another EM1 who spent 10 years ashore. It all goes into how you work out things with your detailer.
As for "fun," you have to decide which rating is fun. If you don't know, go in undesignated in an apprenticeship. There are five - construction, medical, seaman, fireman (engineering), and airman. Then you can strike a rating based on your experience and what you get picked up for. Remember, every job has ups and downs, and in the Navy a bad attitude will certainly magnify your numerous downs.
Speaking of ports, that depends on your platform and homeport. Typically, smaller ships (destroyers, cruisers, and frigates) enjoy more port visits. Amphibious ships and carriers get the tail end. Flag ships get the most (and best) ports because of their VIP status. If you're looking to hit up a bunch of different places, definitely get stationed in Japan. 7th Fleet is underway all the time and going everywhere.
Like I said, enlisted careers are very fluid and limited only by your initiative and aggressiveness with your own training and dealings with your detailer.
Good discussion here::