For the three-dimensional gonihedric Ising models defined by Savvidy and Wegner the bare string tension is zero and the energy of a spin interface depends only on the number of bends and self-intersections, in antithesis to the standard nearest-neighbour three-dimensional Ising action. When the parameter , weighting the self-intersections, is small the model has a first-order transition and when it is larger the transition is continuous. In this paper we investigate the scaling of the renormalized string tension, which is entirely generated by fluctuations, using Monte Carlo simulations for , 0.1, 0.5 and 1.0. The scaling of the string tension allows us to obtain an estimate for the critical exponents and using both finite-size scaling and data collapse for the scaling function. The behaviour of the string tension when the self-avoidance parameter is small also clearly demonstrates the first-order nature of the transition in this case, in contrast to larger values. Direct estimates of are in good agreement with those obtained from the scaling of the string tension. We have also measured .