heycal - to be honest, if you have insurance or can afford it I'd highly recommend seeing a good physiotherapist for a few sessions who can help assess and create an specific exercise program for your back.
Your symptoms (stiffness)and MRI report is suggestive of fairly "normal" wear and tear in the spine rather than a weakness of the musculature where core exercises might benefit. The herniation will most likely heal on its own if you don't aggravate it further.
I'm studying medicine and evidence seems to suggest exercise can prevent or slow progression of your degenerative changes. However, everyone's ailment is different so which exercises to do is the question. Hopefully a good physiotherapist (someone preferably with experience or postgraduate training in musculoskeletal conditions) can: 1) conservatively resolve any acute conditions you have (the disc hernia) and 2) tailor a daily exercise regimen that's specific to you to prevent worsening of the facet joints. You will likely have to practice them daily. A lot of suggestions posted so far may be great, but they may not be great for you specifically. In fact, I can see some suggestions are excellent for muscular back pain which is actually more common than spinal pain.