Split to Dubrovnik

Quickest way is probably a bus - there are numerous buses a day running between towns. 4-5 hours each way I think, depending on which bus it is.

There's also a catamaran which takes longer I think. It may not be quite as scenic as it sounds. The catamarans travel quickly and you can't sit up on deck or anything. I took one from Korcula to Split - it was OK but not necessarily better than the bus. An actual ferry (which would take even longer) where you can go up on deck and watch the scenery would probably be a lot of fun and a smooth ride.

https://getbybus.com/en/
https://www.rome2rio.com/

Define "best".

Cheapest? Fastest? Most fun? Best scenery? Most comfortable seats?

Personally, I enjoy traveling by corporate jet and private helicopter whenever possible - but it's rarely what most people would call "best" (unless you can find someone else to pay for it).

Bus is probably cheapest. Private car is probably fastest and most convenient (other than parking) - having a car is alternately a great convenience and a great headache in Croatia, depending on the location (great for Plitvice, a headache in Dubrovnik). There may or may not be a ferry, depending on the time of year (always check schedules for ferries in Croatia carefully, some don't run year-round, some don't run often in the low season)

What's "best"? It greatly depends on the specific details, the circumstances of your trip, and your preferences/priorities.

We have a party of 3 couples with each having 2 luggage bags. We are looking for the most cost effective relaxed comfortable timely way to get from Split to Dubrovnik. We don’t plan to rent a car or a private vehicle.

Thanks to those that replied already.

I found the Croatian buses comfortable, but I am 5' 4" and don't know what they would be like for an extremely tall person. When I was last in Croatia (2015) it was common to be charged a small fee (roughly the equivalent of $1) for each suitcase stowed in the hold of the bus. You are not being scammed if the driver or an assistant asks you for a bit of extra money.