Apr 2, 6:55am
Hi, welcome to TORn! This is an interesting topic that to some degree we have to speculate about, but we know that Aragorn went to Rhûn, Far Harad, and (it is strongly implied) Mordor. From "The Tale of Aragorn and Arwen":
He rode in the host of the Rohirrim, and fought for the Lord of Gondor by land and by sea; and then in the hour of victory he passed out of the knowledge of Men of the West, and went alone far into the East and deep into the South, exploring the hearts of Men, both evil and good, and uncovering the plots and devices of the servants of Sauron.
"The hour of his victory" refers to his successful raid on Umbar (described in "Gondor and the Heirs of Anarion" within Appendix A), after which:
He sent a message of farewell to Ecthelion, saying: ‘‘Other tasks now call me, lord, and much time and many perils must pass, ere I come again to Gondor, if that be my fate.’’ Though none could guess what those tasks might be, nor what summons he had received, it was known whither he went. For he took boat and crossed over Anduin, and there he said farewell to his companions and went on alone; and when he was last seen his face was towards the Mountains of Shadow.
I can't think of any specific reference to him visiting Khand but I think it's a reasonable supposition that he did. It's the most direct known route to get from the East to the South, in any event. Dorwinion is in my opinion less likely since it doesn't seem to have been under Sauronian influence and also isn't known to have participated in wars against the Easterlings, so there's less of an obvious reason for Aragorn to go there, though that certainly doesn't rule out the possibility. (And it might very well have participated in minor border wars that simply weren't recorded in any of the histories that made it into the Red Book.) Very little is known for sure about Dorwinion, but I'm partial to the theory that it was a Sindarized Elvish realm. Appendix B states that "before the building of the Barad-dûr many of the Sindar passed eastward, and some established realms in the forests far away, where their people were mostly Silvan Elves. Thranduil, king in the north of Greenwood the Great, was one of these." The only other realm we know of for sure is Lórien,* but in my reading the quote implies that there were more than two Sindarin princes who set up realms for themselves in the early Second Age.
Aragorn spent relatively little time in the North so I think him visiting the former Angmar is the least likely of the locations you mention. "The Tale of Aragorn and Arwen" mentions that his mother Gilraen "took leave of Elrond and returned to her own people in Eriador, and lived alone; and she seldom saw her son again, for he spent many years in far countries." While there were threats in Eriador that the Rangers fought against -- eg, "'Strider' I am to one fat man who lives within a day’s march of foes that would freeze his heart, or lay his little town in ruin, if he were not guarded ceaselessly" (FOTR, II 2), which might refer to the Barrow-wights -- Aragorn himself seems to have spent relatively little time engaged in those struggles, being primarily concerned with theatres in which Sauron's direct influence was greater.
*Possibly founded by one Malgalad, who may or may not be the same as Amdir; it's all pretty foggy. And I should note that depending on which version of the history of Edhellond you want to give priority to it might also have been founded by Sindarin settlers around this time, but there does not seem to have been a Nandorin population in the region prior to their arrival since the human fisherfolk of the region fled when the Sindar showed up, and it's not mentioned in the discussion of "The Sindarin Princes of the Silvan Elves" (UT, The History of Galadriel and Celeborn, Appendix B).
(This post was edited by Eldorion on Apr 2, 7:01am)