This methodology is described in the glmnet paper Regularization Paths for Generalized Linear Models via Coordinate Descent. Although the methodology here is for the general case of both $L^1$ and $L^2$ regularization, it should apply to the LASSO (only $L^1$) as well.
The solution for the maximum $\lambda$ is given in section 2.5.
When $\tilde\beta = 0$, we see from (5) that $\tilde\beta_j$ will stay zero if $ \frac{1}{N} | \langle x_j , y \rangle | < \lambda \alpha $. Hence $N \alpha \lambda_{max} = \max_l | \langle x_l , y \rangle |$
That is, we observe that the update rule for beta forces all parameter estimates to zero for $\lambda > \lambda_{max}$ as determined above.
The determination of $\lambda_{min}$ and the number of grid points seems less principled. In glmnet they set $\lambda_{min} = 0.001 * \lambda_{max}$, and then choose a grid of $100$ equally spaced points on the logarithmic scale.
This works well in practice, in my extensive use of glmnet I have never found this grid to be too coarse.
In the LASSO ($L^1$) only case things work better, as the LARS method provides a precise calculation for when the various predictors enter into the model. A true LARS does not do a grid search over $\lambda$, instead producing an exact expression for the solution paths for the coefficients.
Here is a detailed look at the exact calculation of the coefficient paths in the two predictor case.
The case for non-linear models (i.e. logistic, poisson) is more difficult. At a high level, first a quadratic approximation to the loss function is obtained at the initial parameters $\beta = 0$, and then the calculation above is used to determine $\lambda_{max}$. A precise calculation of the parameter paths is not possible in these cases, even when only $L^1$ regularization is provided, so a grid search is the only option.
Sample weights also complicate the situation, the inner products must be replaced in appropriate places with weighted inner products.
