You say "the exponential distribution" and "the binomial distribution", but there are no such things: for each ? > 0 there is a corresponding exponential distribution having that parameter; and for each integer n > 0 and each real p between 0 and 1 there is a corresponding binomial distribution having n and p as its parameters.

That said, the basic idea is to use RandomVariate[dist[param1, param1,...], 1000] to generate a sample of size 1000 from the specified distribution dist with the specified parameters param1, param2, etc. (0 or more such parameters).

To do that 10000 time, you could use Table[..., 10000] and immediately map Mean onto it. For example:

      expMeans = Mean /@ Table[RandomVariate[ExponentialDistribution[1], 1000], 10000]

The mean and standard deviation of those means would then be given by:

     {Mean[expMeans], StandardDeviation[expMeans]}

or both together by:

     Through[{Mean, StandardDeviation}[expMeans]]

And for a histogram, you can either let Mathematica use its default bins or specify them yourself, e.g., by specifying the number of bins, as in:

      Histogram[expMeans, 100]