this thing happens with personal pronouns only. You go verso di me, verso di lei, verso di noi, verso di loro. I can't think of the circumstances in which you can go "verso di te" but someone could throw something, a stone, "verso di te".

Instead you go verso il cavallo, verso piazza San Marco, verso Roma, verso New York and every other thing and living being.

I hope there aren't exceptions. There could be, because what I've just told you doesn't come from a grammar. I'm only trying to remember how I speak and creating a rule in this moment. I'm native, not a teacher.

Never approach: (A) un cavallo from the rear side. (B) un toro from the front side. (C) un idiota from any side

"Vado" means "I go," which doesn't necessarily mean "walk."

'a' is to, while 'verso' is towards. it means almost the same thing, but then again, English has the distinction.

You can say either. Use the former if you want to emphasise that "I" did something.

same thing

I wouldn't say "I go towards . . ." in any situation. It feels regional to me. Grew up in California. I would say "I go toward the (name of animal or other object here)".

Sometimes you can leave "Io" at the beggining to emphasise that it's YOU who's going towards the horse. Other than that, it's unnecessary.

"Towards and toward are prepositions. We can use both forms, but towards is much more common than toward. …"

https://dictionary.cambridge.org/grammar/british-grammar/towards-or-toward