Most people head west from New Bark Town to the next city over — it's called Cherrygrove, and it's a three-day walk from your current location.

As for the Pokemon you ought to get, I'd personally recommend something you'll be able to use for transportation once it hatches. An Onix or a Growlithe, perhaps, though you'd need to evolve the Growlithe before it'd be of a rideable size.