I hope you're not doing this trip in the order presented as it makes a lot more sense to drive San Jose/Monteverde/Arenal/Cahuita/San Jose and will save you significant driving time.
You're covering a lot of ground in 10 days so I wouldn't rely on public buses except perhaps for the last leg back to San Jose, as the buses are definitely slower than shared shuttle buses or driving yourself. No public bus from Monterverde to Arenal. You'd need to transfer buses to get from Arenal to Cahuita making a long day's travel really longggg.
I think whether you drive or not depends on how you feel about driving in a foreign country where signage is not particuarly good and you will have to stop on occasion and ask locals for directions. The only unpaved roads you will encounter are to and from Monteverde which is steep and again around Cahuita which is all dead flat and easy driving. Renting a GPS would help with wayfinding but is not essential if you have a good map.
I've traveled both ways in Costa Rica and liked both options.