Yeah, you do have to grind a bit. Nowhere near as much as some games (looking at you, basically every Final Fantasy game) but the leveling is designed around you doing some extra fights for XP. Every new area generally has a "grind spot" that is moderately to incredibly obvious, typically some grouping of enemies that are enough to fight but not enough to overwhelm you, placed within reasonable walking distance of a bed, hotel, or other way to refill your HP/MP for cheap/free.
For the first town, before you take on the punks roaming the streets you should get some levels fighting crows, dogs, and snakes up near your house. Once you can kill them in two turns or less head into town and try taking on a single punk. If you survive that fight without being nearly dead, keep fighting punks. If you almost die, go heal up and farm a little more. And if you DO die... well you only lose half the money you have on you, so as long as you keep most of it in the ATM you haven't lost much of anything.