How can I most quickly find a big prime, p, for which 4p+1 is also prime? For example, p=37 works. I wonder if these special primes have been researched and some characteristics are known. Are there infinitely many of these primes?



special primes with p'=4p+1
There are very likely infinitely many primes of this form but this is open. 


If one where to count the number of such primes up to $x$, one expects to find 
$$
\frac{C x}{( \log x)^2}
$$ 
for some constant $C$ that one could compute, so on the one hand not too few but still only a set of relative density (in the primes) $0$, as expected by OP and compatible with the observation that early on there are not too few. 


Where does this expectation come from: one can rephrase the problem as the problem of searching solutions *in the primes* of the linear equation $X = 4 Y +1$.


A linear equation (or a system thereof) is *expected to have a infinitely many solutions* in the primes if it has in the integers and there is no *local* restriction, that is there is no congruence relation that 'forbids' all variables to be prime, say $X=Y+3$ cannot have infinitely many solutions as it 'does not work' modulo $2$, *and* there is no problem due to positivity of the primes, say $X+Y = 1000$ cannot have infinitely many solutions in primes.
And, there is also a prediction for the number of solutions.


This circle of ideas goes under the name Generalized Hardy--Littlewood conjectures.


For certain systems it is *known* that there are infinitely many solutions but for others not. Essentially, what is the case (under currently known results) is governed by the *complexity* of the system (*in a precise technical sense* see the reference below). 


The equation you are looking at has infinite complexity (in this sense) and therefore it is not known (yet conjectured) that there are infinitely many solutions. 


The introduction of ['Linear equations in primes'](http://arxiv.org/abs/math/0606088) by Green and Tao gives a good overview; the paper itself makes important progress on these types of problems, note that the results in this paper are formulated conditionally on two conjectures but meawhile these are settled by the same two and Ziegler. 


If you want to search for such primes a thing to note are congruence conditions to exclude unecessary test. For example, one can use that if $p$ and $4p +1$ are prime than $p$ can only be congruent $7$, $13$, $19$ modulo $30$, which follows by looking at the problem modulo $3$ and $5$ (and the info one has modulo $2$). One could add info for additional primes but perhaps this mod $30$ is a good balance. 


