To make use of a resupply ship, you sail "home" to her just like you would your home base. Basically the same rules apply. The game will consider it the end of your current patrol, and when you leave her it will be the beginning of your next one.
I know that the ships that are docked in neutral ports are very easy to find - they are docked, so they're right where the dot on the map says they are, not sailing around so that you have to search for them. If you sail into the port and go up on the bridge and start using the UZO to "ID" all the ships in the harbor (provided you've not gone to all manual targeting, which I assume removes the ID info from your notepad) you will be able to pick her out that way. Or you can get plenty close by zooming in on the NavMap and just sailing right up to her.
You don't have to manually pull your boat up alongside (but you can if you want), you just have to get within the same distance from her location as you would the "dot" that represents your home base when you're docking there. Once you're that close you do the same thing you would do when returning home - hit Esc, select Exit Patrol, and then "Dock at [name of ship]."
The game will show you as having ended your current patrol there. Then you'll start your next patrol from the same location.
If you are using SH3 Commander, it will automatically start the next patrol 1 day later. I don't know if the game by itself does that or if it follows the same "rules" it uses for time spent in your home base between patrols. I suspect it follows the normal unmodded rules, otherwise Commander wouldn't be set up to compensate for it. Either way, though, the game will automatically add extra time in between patrols for any hull damage you've taken. That can be manually edited out of your career files in between patrols if it's a big deal to you, otherwise, don't worry about it. Unless you're trying to keep ultra-realistic patrol logs when it comes to such things, it's not really a gamebreaker.
The ships that are not docked in some port may be sailing around the general area, or may be at anchor in the designated spot - I really don't know! But I wouldn't think they'd be too hard to find. I expect if you get as close as possible to the dot where they're supposed to be they'd come into visual range (or at least hydrophone range) sooner or later.
Also, be aware that you will not have the same "between patrol" options at a resupply ship that you have at home base. You get food and ammo and hull integrity is restored (and other damage repaired), but you can't upgrade equipment or choose a different torpedo loadout. You'll just have to take what they've got to give you. And obviously you won't be able to dismiss/recruit crewmen, stuff like that.