I'm the 11:38 poster. I've been to Costa Rica several times but I don't have car rental tips or hotel recommendations as I stay with friends who live there (and who drive us all around - or we all take the bus). I imagine that GPS would be very helpful for driving, though.
As for other things to do, Inbioparque on the outskirts of San Jose is worth checking out. It's a good educational introduction to the ecosystems of Costa Rica. We had a lot of fun there as adults and my friend's children (young elementary school age) really enjoyed it as well. You can also use it as a stop on your way out of town to the other destinations.
Enjoy! Costa Rica is a wonderful place to visit!!!

Not OP but planning a similar trip this summer with kids of similar ages.

Any car rental tips? Hotel recs? Must do or do nots?

La Fortuna is sort of the "gateway" to Arenal. The town is a bit of a tourist trap, but there are a lot of accommodation options, so you should be able to find something to fit your budget. As far as activities, check out the hanging bridges (for the self-guided tour, your kids would be free!). I've seen all sorts of cool animals (including monkeys) there. There are a few hot springs resorts around La Fortuna where you can pay for day admissions and get access to the pools, slides, etc. That would likely be more cost effective than staying at one of the resorts. If the kids are into snakes and/or butterflies, the EcoZoo is also very cool. Enjoy your trip! Costa Rica is amazing.

We're planning a trip to Costa Rica and have a hole in our itinerary we're trying to fill, so I would love any suggestions. We already have plans for the beach, but are interested in spending 2-3 days at Arenal, Monteverde, or really anywhere else fun and not too far from San Jose. We're on a pretty tight budget, so the less expensive the better (we don't mind rustic accommodations) but I wouldn't mind splurging for something really great. The kids are a bit young, and timid, for the high adventure stuff, but we would love to see cool animals (especially monkeys!), volcanos, hot springs, waterfalls, etc.