(EXTREMELY LONG ANSWER WARNING)

god I am so extremely into this question. I sort of like to break his early wandering up into a few stages!

I think a lot of his very immediate wandering was pretty defined by just complete confusion- he was never given an explanation in any way of his state of being. Learning things like "you are invisible to all things" and "nobody can hear you" and "you cannot touch, interact with, or even move normally through the world" would be very significant learning curves and I think that some amount of time would be spent in a very scattered, confused state of trying to understand what /being/ would mean from now on. Not being able to really walk (if he can't touch anything, he can't... walk on the ground...) and not yet understanding what physical space meant to someone who isn't a part of it anymore meant he didn't wander at first so much as unintentionally End Up places, in a strange slideshow of locations that he struggled to control the flow of.

There would also be the adjustment of time- or a lack thereof, with the Flood stopping the passage of days. This, combined with his now invisible state meant that time essentially meant nothing at all; things just happened and stopped happening and there was no indication of other change to help along processing or mental reset. (This is still an issue, but having access to the Source's day/night cycle and the ability to go dormant as Roku's rejoined shard offers what is somewhat like a sleep state, to his mental state's great relief.)

So- at first, I think his actions were a lot of him just standing places. Disconnected, unintentional places- locations he thought about and was suddenly at, without meaning to have moved at all. Very much the "man standing beside a light post who's gone after a car passes by" sort of energy- observing in mute silence before he's gone again, spirited away to some other place he once knew. I affectionately think of this as the shock stage. He cannot even start to think about what's going on; it's too overwhelming, and he narrows down his scope to what he CAN control.

Learning to pretend to walk and drift more intentionally was an early focus just to give himself some semblance of understanding and control, which began his wandering in earnest. With some grounding in reality, even if it was pretend, he could even start processing what he was actually seeing as His World, even if he was no longer a part of it. This moves into the anger / FIGHT stage, defined by seeing Too Much of what has become of the world. People are not just suffering, they're being slaughtered. The moments after the halting of the Flood are filled with entire towns being wiped out and cities being abandoned entirely; try or not, he's present for too many of these moments, and his instinct is that something needs to be done and he needs to fight back, and he cannot. There's a LOT more of the SHB style "trying to do something and nothing happens" resistance around this time, where he cannot accept that he can't do anything, and has to be forcibly reminded time and time again by being the last one standing in towns left silent.

If you've seen Godzilla Minus One, I think that the survivor's guilt story there is a VERY accurate representation of pretty much the rest of his descent, aside from the uh- part where the main character starts healing. He does not do that. Grief and guilt are the soon-following neighbors of anger, when forced at last to understand that this is his fate. If he had been wandering before, now it's desperate and senseless. Walking but also fraying between here and there- following the edge of the Flood wall until he's seen every inch of its parameter, seeing too many familiar routes and towns cut in half or left in ruins, and ever aware of the crushing lack of passing time or possible End Point of it all.

This is, I think, where the fraying starts happening- there are only so many times you can be the last one standing in a town you once loved, in the deafening silence after a single Sin Eater wiped out every last living thing. Similarly, there are only so many times you can hear survivors desperately trying to find any reason that this could be happening to them, and find blame in the obvious culprits of himself and his friends. The complete isolation, lack of passing of time, and lack of any sort of reprieve (he can't even sleep) would be absolutely unbearable for the most mentally sound person around and considering all that had gone down before his death, he would hardly be in a strong state of mind! Which turns into fraying, disassociation so strong his soul starts coming apart at the seams. But this is a very slow process, and at first, in those early post-Flood times, manifests mostly as a sort of deepening mental rut; the repeated questioning of why he's here, why he was left behind, why did this happen, why did he do this, etc etc.

This early time is also probably where most like... actual breakdowns or emotional outbursts might have happened. It's an unbearable state of existence, and before the fraying gets severe enough that he can't gather his thoughts enough to form a reaction, these are the times when he's most likely to vocally plead with Hydaelyn or actively show grief. It's through spending everything he has on fruitless efforts and his upset being met with the same agonizing silence that he mellows out into the Ardbert of ShB- like, what's the point? What difference will anything make? It's now a necessarily friendly mellow, just a deadened one. Early ShB Ardbert just wants to die, and makes that terribly clear. (this is the Apathy Stage. he is not actually apathetic, but like please god just let him die to escape this punishment. a selfish thing to ask, when faced with the suffering of his world.)

AS FAR AS his actual actions- at first he goes to check on people he knows! Towns, familiar faces, etc- keeping tabs on what he remembers, especially as time drifts strangely on and people age and he has no way to comprehend time beyond that. This is, however, something he stops doing pretty fast, and starts retreating to places where he won't see people up close; it turns out to be deeply self-destructive to linger too close. This helps lead into his SHB behavior of watching from the sidelines and keeping his distance, even fully aware that he's invisible. He has been removed from the narrative; best not to fight that by getting too close.