"How do I find out about all his trips? Would I to visit the NA in person and search myself?"
Before actually visiting the NA it's best to search online to see if they have the actual documents you're after.
To be frank, I've never had any ancestors in the merchant navy so I really don't know what sort of records are kept (the nearest I've got is some ancestors who used to work on canal barges).
I've had ancestors who were in the navy and there are records of each ship and shore station where they served, but I have no idea what records exist for the merchant navy.
The first step would be to search online at the NA. Here are a couple of suggestions (which you may already have searched):
https://www.nationalarchives.gov.uk/help-with-your-research/research-guides/merchant-navy-ships-records-crew-lists-musters-and-log-books/
or here
https://www.nationalarchives.gov.uk/help-with-your-research/research-guides/merchant-seamen-serving-since-1918/
Some records are available to download online. There is a charge of £3.50 to download these, but, if you register with the NA then they are free of charge.
Sometimes, records are not available to download. The only reason to actually visit the NA is if you want to see a document that is not available online.
Below is an image of the message you get when a document isn't available online. If you see this message then you have to go to Kew to see the document.
"I think you have to book a timeslot for Kew as well which will need some planning if not close."
No, you just need to book a particular day and you can turn up anytime on that day you like.
If you've never been to Kew before and you want to view any documents you first have to register for a Reader's Ticket online which you do when you book a visit:
https://www.nationalarchives.gov.uk/book-a-reading-room-visit/
So, you say what date you want to visit and then provide a list of what documents you want to see.
They also have desks with camera stands. So you just screw your camera to the stand facing down and then instead of having to take notes or anything you can quickly and easily take a photo of every page of the documents that you are looking at.
This makes things so much easier but you do need to specifically request a desk with a camera stand.
The NA can be a bit of a trek to get to. I come into London on the train into Kings Cross and from there take the tube to Kew Gardens. It's then about a 10 min walk to the NA.
When you arrive there is a cafe there but it is horribly overpriced and not very nice.
You then put all the stuff you're not allowed to take into a locker:
https://www.nationalarchives.gov.uk/about/visit-us/researching-here/can-take-reading-rooms/
and go upstairs. If you haven't been before then they will take your photo and print out your Reader's Ticket.
You then go into the reading room and head to the desk. They will typically have your documents waiting for you in a locker. You then go to that locker and take the document or documents and then go to the desk that you have been assigned.
You then take your photos or anything else that you want to do and then, when you're finished, you return the documents to a different desk.
But the thing is, unless you can find the documents you're looking for on the NA website then going there in person won't help you at all. You really do need to identify what documents you're interested in and for them not to be available to download.