No, there is no need to purchase any equipment until you have had your initial appointment with your trainer. At that time, the trainer will advise you as to what types of equipment to buy. Often, the trainer will lend you equipment to try out for the first few weeks and then can tell you how to order the equipment.
Examples of equipment the trainer may ask you to use are physio balls, medicine balls, free weights, elastic tubing and jump ropes. Very little space is actually needed to train at home. A 10 x 10 area free of furniture and other obstacles is all that is needed.