Finding strong primes Wikipedia lists the following conditions for a prime to be strong:

$p-1$ has large prime factors. That is, $p = a_1 q_1 + 1$ for some integer $a_1$ and large prime $q_1$.

$q_1-1$ has large prime factors. That is, $q_1 = a_2 q_2 + 1$ for some integer $a_2$ and large prime $q_2$.

$p+1$ has large prime factors. That is, $p = a_3 q_3 - 1$ for some integer $a_3$ and large prime $q_3$.

This seems pretty vague to me. What does "large prime" mean? How large should $p_1$ be compared to $p$?

And, most important, what is the standard way to generate large strong primes, or to check if a large prime is strong? Also, is using safe primes (of the form $p = 2*q + 1$, where $q$ is a prime) safe and/or efficient?
All the properties discussed in the question are for strong primes $p$ in the context of using $p$ as a secret factor of a large composite $n$ which factorization should be intractable; these properties are given in the landmark RSA paper (1978), without justification. Properties thought for (often public) primes used in other cryptosystems can be different.

The first property


  $p-1$ has large prime factors. That is, $p = a_1 q_1 + 1$ for some integer $a_1$ and large prime $q_1$.


is intended to make factorization of $n$ using Polard's p-1 algorithm hard; this is relevant to some degree because this algorithm has a component of its cost proportional to the highest factor of $p-1$, where $p$ is the factor of $n$ exhibited by the algorithm.

Similarly, the third property


  $p+1$ has large prime factors. That is, $p = a_3 q_3 - 1$ for some integer $a_3$ and large prime $q_3$.


is intended to make factorization of $n$ using Williams' p+1 algorithm hard; this is relevant to some degree because this algorithm has a component of its cost proportional to the highest factor of $p+1$, where $p$ is the factor of $n$ exhibited by the algorithm.

One standard that considers the above is FIPS 186-4 (appendix B.3), with $q_1$ and $q_3$ required to be larger than 100 bits when $p$ is 512 bits (and optionally: $q_1$ and $q_3$ larger than 140 or 170 bits when $p$ is 1024 or 1536 bits). It is hard to tell quantitatively when it makes actual sense taking precautions regarding such properties  (beyond conformance to standards): it is pointless when generating a few two-factors RSA keys of at least 1024 bits; it might be useful when extremely many RSA moduli with relatively small factors (e.g. in multi-prime RSA) are generated, and an adversary would benefit finding the factorization of any modulus (rather than of a particular modulus); see this question and its answers.



The second property


  $q_1-1$ has large prime factors. That is, $q_1 = a_2 q_2 + 1$ for some integer $a_2$ and large prime $q_2$.


has to do with reducing the odds that an RSA cycling attack succeeds; see this answer. I fail to find a quantitative assessment of how big $q_2$ should be, or any modern standard with this requirement. I accept the consensus that RSA cycling attacks have negligible success odds for practical key sizes (but would very much appreciate a quantitative argument!).



The standard ways to generate a large strong prime $p$ with the first and third properties (perhaps the second if required) is to choose the auxiliary primes $q_2$ (if required) then $q_1$, and $q_3$, then generate $p$; rather than choosing $p$, then trying to exhibit $q_1$ (then $q_2$ if required) and $q_3$. The later approach would be very computationally intensive: if $p$ is a random 512-bit prime, it requires the factorization of a $(p-1)/2$ and $(p+1)/2$, which are mostly random 511-bit integers, and that's typically a hard job. The former approach (used in practice) introduce significant complexity, but relatively little computational burden; several such methods are detailed in the aforementioned FIPS 186-4 appendix B.3.