I rented an 8 seater minivan from the falmouth port a few years ago. It was very easy. I think it was from island Car Rental or something like that? It was cheaper than the big names. They're right in the port area (amongst the shops). We just headed straight there after we got off the ship. It did take a bbit longer than a typical US rental car office - maybe about 30+ mins? No big deal. It wasn't as slick as it is in the US, but we were completely safe and did not deal with any upselling.
I chose to use a credit card that covered insurance in Jamaica (I read the fine print carefully) and did not have to pay extra. It was very inexpensive - cheaper than paying a private tour guide (which was the initial plan).
Driving to Ocho Rios was easy - there is a coastal highway (that was in great condition when we drove it a few years ago). We did Dunns River Falls by ourselves and it was sooo much better than joining those tour groups with the endless line of people holding hands. We paid $20/pp for entry and got to climb it at our pace, stopping whenever we felt like it to soak in a pool or under a waterfall.
On the way back, we stopped at a roadside stand to buy a $1 coconut (it was funny since the guy had a machete - we sent my DH out by himself while we locked the doors haha!). Once we felt safer, my girlfriend went out to take photos. We also stopped at a restaurant off the side of the highway - we chose one that looked larger and cleaner across from small tourist hotels with a bunch of tourists around. It was fun to have some local jerk chicken.
We saved a bundle renting a car and gained a lot of freedom. Friends were freaked out when we said we were going to drive in Jamaica - but frankly, the roads were good and we were perfectly safe the whole time. You do need to stay aware and take some extra safety precautions. All of us have also driven in 3rd World countries and I personally did not find Jamaica that much different.