Best or recommended R package for logit and probit regression
Unless you have some very specific or exotic requirements, in order to perform logistic (logit and probit) regression analysis in R, you can use standard (built-in and loaded by default) stats package. In particular, you can use glm() function, as shown in the following nice tutorials from UCLA: logit in R tutorial and probit in R tutorial.

If you are interested in multinomial logistic regression, this UCLA tutorial might be helpful (you can use glm() or packages, such as glmnet or mlogit). For the above-mentioned very specific or exotic requirements, many other R packages are available, for example logistf (http://cran.r-project.org/web/packages/logistf) or elrm (http://cran.r-project.org/web/packages/elrm).

I also recommend another nice tutorial on GLMs from Princeton University (by Germán Rodríguez), which discusses some modeling aspects, not addressed in the UCLA materials, in particular updating models and model selection.