Ranged combat in Monster Hunter is more expensive than using the close-combat weapons, since you have to create separate "gunner" armour sets and buy/make the ammo for bowguns. The gunner armour also has less defence of course so you'll also have to make sure you stay out of the way of attacks. You don't need to make arrows for bows and you get an infinite amount with them, but they do use "coatings" and it definitely helps to buy/make these. It's also more ideal than with other weapons to have various different types of bows depending on the monster you're fighting.
The bow would probably be better to start with than the bowguns and less expensive, but the close-combat weapons are easier starters. Bowgunning is best after you've amassed a lot of wealth and items and have nothing better to spend it all on.