one and a half and 3 and a half years old? Im not sure the one and a half year old will take in much of the scenery and long drives will definitely be uncomfy .
i would NOT suggest driving long distances then of course. So avoid all the cross island highways, they are uncomfy for many ADULTS. thousands of turns , etc. They are really only good for the driver (everyone else can get car sick).
I would suggest yangmingshan. YOu could drive up to yangmingshan and there is a little park there to take a break and also a nice western restaurant too. Plus there is a nice hotel there you could stay overnight at and if its summertime a great pool (cool water tho). And then drive on down to GIN SHAN . Its a pleasant road and has a lot of places you could stop for awhile. Theres a fumarole (its a volcano thats dormant, but not dead) and you could get to the top of 7 star mountain and check out the scenery from the nearly 4000 feet high peak. And then when you get to ginshan you could turn right and then visit YEH LIU, with its little park, great place to see the funny shaped rocks and also visit the OCEAN PARK there, They have dolfin shows .
and then you could either continue on to KEELUNG to see the busy small harbor. And drive up to the park there that has that budda. and ur kids can use the bumper cars there.
OR you could go back towards gin shan and continue on the scenic road running alongside the ocean to tamshui and theres a cute place to walk around and the kids will like it.
Then take a side triip by the NEW tilting train to taroko. If you have time, try to make a stop in Nan fan ao , (buy two tickets on the train. one to nanfang ao and then hours later onward to hualien) . Its probably taiwans largest fishing port and worth a visit to the seafood restaurants there to eat live steam crab, fresh fish, etc. pretty scenic.
spend the night in hualien and take a cab (dont even need to rent a car) to taroko and enjoy the gorge . Hualian has a very unique and nice feel bout the place. Its hard to describe, but its definitely a VIBE about it.
Then continue by train down to Taitung. And if you can stay at the ROYAL CHIHPEN hotel. at least one night. Beautiful place in its own environment. Great relaxing place for kids and adults.
and then take the train again down to the southern tip of the island . And then spend a few days there at the southern tip of the island. YOu can rent a jeep to drive the family around. And you can go swimming in beautiful waters. and rent a jet ski and all that.
stay at the howard plaza hotel there (it has a water park indoors , great for kids) or next door at the ceasars park. lovely place, very relaxing and kids will love the swimming pool, the tennis courts are often free of people too. and the wondeful beach tehre is just awesome. clean and beautiful.
and then take one of the many buses onwards to kaohsiung and enjoy the sights of a major taiwan city. Btu one thats also taiwans biggest harbor. And there is a park shi chi wan. that you can go to and walk along the shore there. And there is a ferry , small boat to go across the harbor entrance to the little island there, that has a lot of seafood restaurants. Kaohsiung has a cute zoo too that kids will like.
And on the way up the island. this time on the west side. be sure to try the new highspeed train. kids and you will enjoy taiwans latest toy , the newly finished 15 billiondollar train. Take that to chiayi, Stay overnight in chiayi and the next day rent a car to drive the two hours up to alishan. OR better yet. take the tiny train up to alishan. spend the day there in taiwans high mountains and take the train back. the train takes 3 and ahalf hours each way i think? but nice.
goes thru something like 49 tunnels and is memorable (I took it only once and i was a kid then and i STILL remember it).
if you can , stay anight at alishan and get up at 430 in teh morning for a short walk to see the sunrise. again something a kid will remember for long time (even if you thought it was just OK).
lots to do in taiwan, even with kids
And of course you shoudl visit the beautiful sunmoon lake and stay a night there. the lake has boats that will take you around. or you could rent your own rowboat and just row around the beautiful blue/green waters .
theres more , but that willll do for now? let others give more ideas.