What exactly was the pirates' plan?

There are two sources of income for Somalian (and other contemporary) pirates: Valuable goods on board and ransom. Of course the valuable goods on board, typically the seaman's pay, is not enough to fund long running pirate activities. It may be enough for single actions but in the movie the pirates had an organisation and backers behind them. Even when it is only for the wages this can be planned, e.g. some pirates get the information about the date of the pay day and attack on that day.

Normally and in this case it is simply about ransom. They have the crew, they have a ship that may be worth 100 million dollars, they have a cargo that may be around the same value. They can't directly cash this but for the shipping company paying a few hundred thousand dollar still is economically more sensible than losing this completely. Military presence doesn't work very well normally, it is too slow and too unreliable from the economical perspective. In the long run secure shipping is better, in the short run and for the single actor (like the shipping company) paying the terrorists / pirates is the cheaper alternative most of the time. The depiction of the military was a tad bit too positive, cheesy, and simplistic in this movie. In fact it looked a little bit like a Top Gun on the water recruiting video at the end.

As seen in the movie there were backers who probably were "responsible" for this part, it wasn't a spontaneous attack. Often the ransom money will be simply dropped in bags out of planes. But that part wasn't explained in the movie; I don't know about the book (which is said by other crew members to be a little bit too positive about the role of Captain Phillips... who wrote it himself).

Wikipedia has an article specifically about Somalian pirate activities: Piracy in Somalia

You're probably correct that the Somalis' intent was to secure the ship and ransom it.

Remember that the pirates were successful only on their second attempt, after an extended chase that put them far outside the original range of capture. Their first (failed) attempt involved two teams and a "mothership" on standby, providing more manpower if nothing else. That first plan may have involved a further network who could tow the captured ship to shore, and "upper management" was within range to dictate action, transport the hostages elsewhere, et cetera. So much goes wrong during the narrative that we tend to see all decisions as acts of unplanned desperation after a point, but we can still guess at their original intentions if when they try to tow the Maersk with their lifeboat, they were following the same playbook as they started out with two teams and the mothership. It's just moot by then.

Speaking of narrative, the film is based on the questionable personal account of Richard Phillips, and further creative license may have been taken by director Greengrass. Neither of them are in a position to speak the minds of the real life pirates who attacked the Maersk Alabama, and may have deliberately left it vague in the storytelling.

The 2012 film A Hijacking (original Danish: Kapringen) is similarly about a commercial ship captured and ransomed by pirates, and painstakingly covers the negotiation process that concludes in a large payout, suggesting that pirates in general are driven by this goal.

Author Jay Bahadur wrote an illuminating article for The Guardian in 2011 about the lives of Somali pirates, the motivations for hijacking commercial ships, chain of command and ransom payouts, and so on.

Relevant excerpts:

... The commercial ships, identifiable by the cranes visible on their decks, were much slower and easier to capture.

... Instances of the crew fighting back were rare, and rarely effective, and the whole process, from spotting to capturing, took at most 30 minutes.

... The captured ship was then steered to a friendly port – in Boyah's case, Eyl – where guards and interpreters were brought from the shore to look after the hostages during the ransom negotiation.

First of all, I'm not sure the Navy's presence really worked as a definite deterrent that would stop the pirates from their determined plans, even if they realistically wouldn't have had a chance against them. Even more than that, once they had the ship under control, they would have hostages and the Navy, though surely able, would certainly think twice before entering the ship. And in addition to that, private organizations like the shipping company, are IMHO more likely to pay the ransom than governments, be it only for the publicity of not letting die their employees for mere "closefistedness".

So I don't think the plan of keeping the whole ship for ransom is that implausible. Other than that I'm also not sure their initial plan wasn't to capture the captain for ransom. So I think both those plans were not entirely implausible, even if I, like you, am not completely sure if the capturing of Captain Phillips was only a short-circuited action or not.