There are basic equipment requirements for some training programs, but with these basic tools, you'll be able to train effectively for years to come.
To train efficiently in the movement patterns that carry over to surfing, applied resistance is required. Take paddling, for example, it's a pulling movement, so various forms of pulling need to be trained to develop endurance and durability.
A set of Olympic Rings, elastic resistance bands, or some light kettlebells will allow for a vast array of pulling exercises to ramp up your paddling endurance.
Much of the Rotational Core work requires either an Exercise Ball or Resistance Bands. Developing strength and power for the lower body requires the addition of resistance so a Kettlebell becomes quite beneficial.
Training with weights or resistance won't make you into a meathead or overly bulky. What it will do is help you to develop a stronger and more capable body.
If you don't have all the equipment right away, no worries. For nearly all exercises there are alternative movement options if you don't have that particular piece of equipment.