Personally I favor solo travel in Pokemon RP's because unless we have a deep seeded plot, it's just going to be a bunch of fluff with no action for the most part. I'm a story writer, so I plan to put Perdy inside situations that others are free to add themselves into if they please, but if not it'll be his own thing. I tend to write longer posts, so some of his shorter misadventures will be single-post anyhow, but there will probably be times Perdy will fall behind. But I'll manage to catch him back up to the others for the end if we reach there.
Of course, depending on what happens, Perdy might be more determined to just move ahead for the sake of managing to beat BRIAR, or if he's chasing tail for the sake of impressing someone. (And failing. Horribly, I might guess.)
I'd also be interested in contests, like
@Leoven. But contests might be more spaced out than gyms, so they could be juggled if one desired it. (Perdy would maybe challenge a gym here and there, if he's given motive.)
Don't take this as me being some loner, however. I
love character interaction. I just hate it when I spend a bunch of posts not doing anything when something could be going on. There are times where this is fitting though, like if there's a huge storm and everyone has to hold up in a Pokemon Center for the night, it'd give a good break to the group and let them talk (or fight) among eachother.
Sorry this was kind of long, just wanted to give my 2cents.