I have a random variable, X, with cdf $F$ and pdf $f$. I want to estimate a parameter of $F$, say mean, $\mu$.
So what do I do? I construct an estimator, $Y_n$, with several random variable X1, X2,..., Xn from $F$ such that,
$Y_n=g(X1, X2, ..., Xn)$
and make sure that the random variable $Y_n$ has the property, $E[Y_n] = \mu$.
Now what? Initially I had problem 1 and now I am stuck with problem 2. How will I find $E[Y_n]$?
Example:
let $Y_n = 1/n*\sum_{i} X_i$ and we do have $E[Y_n] = \mu$. How do we find $E[Y_n]$?
Suppose we take single observations {x1, x2, ... xn} from our iid $X_i$. How will $y_n = 1/n*\sum_{i} x_i$ approximate to $\mu$? I know $\lim_{n \to \infty} Y_n$ converges in probability to $\mu$. But that does not mean the value $\lim_{n \to \infty} y_n = \mu$.
Where am I wrong?
I am not sure this answers all your questions. Maybe it helps to start by understanding how much knowledge we have about the objects involved. In the classical setting of statistical estimation you give, $\mu=E(X_1)$ is a fixed number which we do not know. But we will assume here that it is finite for simplicity. We do not "find" it like we find the solution of an equation (if we could do that, we would not need statistics), but we can construct estimators for it.
Assuming we have a sample of independent realizations $X_1,\ldots,X_n$ of $X$ and defining $Y_n=(X_1+\cdots+X_n)/n$ as you did above, we get a random variable $Y_n$, which we use to estimate $\mu$. As you stated above, $E(Y_n)=\mu$. But that is not the point, also $E(X_1)=\mu$. So why is it better (not always, only on average!) to use $Y_n$ rather than $X_1$ to estimate $\mu$? Assuming $\sigma^2=\mathrm{Var}(X_1)<\infty$, we find that $Y_n$ has a smaller variance than $X_1$ (namely $\sigma^2/n$, compared to $\sigma^2$). The average squared distance from $Y_n$ to $\mu$ is $n$ times smaller than the one we get if we use $X_1$ as estimator for $\mu$.
We can compare properties (like the variance) of different estimators, because as they are random and may give different results for every sample, we never know for sure what $\mu$ is (in a finite sample). But we may be able to compare e.g. how much the estimates fluctuate around the true value.