Car or Bus - Amsterdam to Bruges with stops along the way

How many days do you have planned to do this? And when exactly? What does your day-to-day itinerary look like?
Assuming that you will fly, have you considered the possibility to fly open jaw; into Amsterdam and back out from Brussels? This will save you a lot of backtracking.

I would take the train. It's very easy to travel in and around the Netherlands and Belgium by train, and the places you want to visit have good public transport within the city, and are also very walkable. Having a car would be a liability in my opinion. Plus you have the extra fee that is usually incurred for taking the car out of the country. That fee gets even higher if you pick up a car in one country, then drop it off in another (that's if you use the multi-city option Dutch_Traveler is suggesting.)

And regarding the multi-city option, I would just stick with flying in and out of Amsterdam. Bruges is only 3 hours from Amsterdam by train so It really doesn't make sense to fly out of Brussels; especially since you would spend an hour getting there from Bruges. AMS is also such an easy airport to fly in and out of, and there is a much better chance of getting a non-stop flight out of AMS. I doubt very much if that would happen with Brussels.

We were just in Belgium and the Netherlands. Their train systems are exceptional. I'm with Mardee, I'd use the trains. If you have interest in spending a night or two in Brussels, then perhaps fly out of Brussels. I really enjoy staying in city centers. I especially enjoyed staying in the charming town centers in Bruges and Delft. I would not want a car in either city.

Will you fly into Amsterdam’s Schiphol airport? If so, take bus 361 from Schiphol to the Lisse Centrum station (30m). Keukenhof Gardens is a short taxi ride or a 30m walk. To get from Lisse to Amsterdam take bus 162 from Lisse Centrum towards Hoofddorp to Nieuw-Vennep, Pondweg (15m) then hop on the train to Amsterdam Centraal (Amsterdam’s central station): https://www.rome2rio.com/.
From Amsterdam Centraal take a direct train to Delft (1h): www.ns.nl. From Delft take a direct train to Rotterdam Centraal (15m). From Rotterdam Centraal take a train to Bruges that involves a transfer in Brussels (3h 15m).

I’ve read your edited post with your more detailed itinerary and you definitely don’t need a car for any of it. You will have a far more relaxed and enjoyable experience if you use the excellent public transport system we have here.
It’s sometimes difficult to understand for those who come from a country where the Car is King, but things are very different here. Cities take extreme measures to prevent people from driving their car into the city centers. These measures range from Low Emission Zones for which you will need to register your car, to pedestrianised streets to parking costs of €10 or more per hour. Scheveningen even introduced a fixed parking fee of €50 regardless if you park there for just 15 minutes or a full day.
I urge you to reconsider and dump the car.

@MaryPat; you seem to have mixed up the public transport companies. Trains in the Netherlands are operated by NS, this includes the train from Amsterdam to Delft. GVB operates busses, trams and metros in Amsterdam and immediate surrounding area.

Rome2Rio is a 3rd party ticket reseller. There is no need to use their service and pay extra for your ticket.
Train tickets in the Netherlands can be bought at the train station, on the website of NS, in their app or by simply tapping in and out with your credit card, phone or smartwatch.
For train schedules and fares, see www.ns.nl/en
For more information about tapping in and out with your credit card which is available on all public transport in the entire country, see www.ovpay.nl/en

Yes, completely possible by train, and recommended.

re. Rome2Rio, I would never buy tickets thru them, but, they are a helpful first step for those new to an area/country to scope out possible transportation options--car, train, bus, walk, etc. From there I would go to the company/country specific sites to get exact schedules and to buy tickets.

Sometimes I am actually surprised by Rome2Rio. A few years ago, my daughter was coming home for Christmas from Honolulu and needed to get to the airport. It was after bus business hours and the company's system was down anyway. I believe google maps gets their info live from bus/train companies. Therefore, google maps was unable to provide a route. It was during COVID so UBER/LYFT wasn't as available. I gave Rome2 Rio a try, and they actually had the correct routes, bus numbers and times.

So I wouldn't bet my life on Rome2Rio, but I don't completely ignore them either.

Month and year?

It's difficult to express how difficult it will be to drive into Bruges or Amsterdam with a private car, and even harder (and expensive) to park. The entire drive north of Brussels, to Bruges will be like Saturday mall traffic in the United States. You may also be surprised by how narrow the garage aisles are, and how small the parking stalls are. I've only been in Rotterdam briefly, but I doubt the car is an asset there, either. Can you drive a Standard if they don't come up with your Automatic?

Although there are many temporary, special bus services for Keukenhof's six-week or so season, you may want to make sure that a bus is better elsewhere than are the excellent trains in these two countries. While Rotterdam or The Hague are okay for overnight stays (you'll be visiting The Hague from Schveningen if the weather is bad), it's possible (i.e. I have done so) to visit Delft, The Hague, Leiden, Edam, Gouda, Alkmaar, and a few other towns entirely by cheap, unreserved local trains, while sleeping in Amsterdam.

We went to Lisse for a once-in-ten-years garden show. Why are you going there?

I'm glad you have five nights in Amsterdam, it's certainly worth it.