I am running Red Hat 7 and I would like to update my TLS / SSL versions to the latest. I'm new at running a web server so my questions are..

How do I check which version of TLS / SSL are currently installed?
How do I update them?

Thanks so much for your help!

This is the version of OpenSSL I am running. How can I tell which version of TLS / SSL is currently being used? I don't want to update just yet before I figure that out.

Thanks for your help so far!

OpenSSL 1.0.1e-fips 11 Feb 2013
built on: Fri Apr 29 08:57:57 EDT 2016
platform: linux-x86_64
options: bn(64,64) md2(int) rc4(16x,int) des(idx,cisc,16,int) idea(int) blowfis h(idx)
compiler: gcc -fPIC -DOPENSSL_PIC -DZLIB -DOPENSSL_THREADS -D_REENTRANT -DDSO_DL FCN -DHAVE_DLFCN_H -DKRB5_MIT -m64 -DL_ENDIAN -DTERMIO -Wall -O2 -g -pipe -Wall -Wp,-D_FORTIFY_SOURCE=2 -fexceptions -fstack-protector-strong --param=ssp-buffer -size=4 -grecord-gcc-switches -m64 -mtune=generic -Wa,--noexecstack -DPURIFY - DOPENSSL_IA32_SSE2 -DOPENSSL_BN_ASM_MONT -DOPENSSL_BN_ASM_MONT5 -DOPENSSL_BN_ASM _GF2m -DSHA1_ASM -DSHA256_ASM -DSHA512_ASM -DMD5_ASM -DAES_ASM -DVPAES_ASM -DBSA ES_ASM -DWHIRLPOOL_ASM -DGHASH_ASM
OPENSSLDIR: "/etc/pki/tls"

That version of OpenSSL will support up to TLS1.2

As for "what's currently being used", that'll depend on where it's being used and what it's configured to use by the application.

If you're using it on a website then run your website through https://www.ssllabs.com/ssltest/index.html which will give you a comprehensive breakdown of protocols and ciphers.