Taitung (台東， Tai2dong1) is a city in Taiwan.
North towards Hualien
There are three main roads that lead north out of Taitung. The one you're probably going to take is number 11, which goes along the coast and thus has the best views.
It's super easy to north from on or just before the bridge that's just to the north of Flowing Lake (google maps link). You can walk there from the city centre.
There's a reasonable amount of local traffic but limited traffic going all the way to Hualien. That said, you should still find it easy to get a lift. We're talking about wait times of more like 15-20 minutes rather than 2 to 5 for locations in Taiwan with more traffic. It may be worth having a sign saying Hualien (花蓮）as well as one with the names of a few local towns (you can see these on google maps).
On the way north you might like to check out some locations like Sansiantai Arch Bridge 三仙台跨海步橋 which is pretty spectacular!
Traffinc to the East of Taiwan from the West is a lot less frequent than traffic between cities on the West coast (e.g., between Kaohsiung and Kenting). This means that it'll be easiest for you to get a lift to Taitung by using two signs. For example, if you're going south from Tainan or Kaohsiung to Taitung, you should have one sign that says Kenting (墾丁) and one that says Taitung (台東). Similarly, if you're going north from Kenting you should have a sign that says Kaohsiung (高雄）and one that says Taitung (台東). That way, anyone going between the two cities can pick you up, and from there you can be asked to be dropped off at Fenggang.
Fenggang is a key transit point from West to East coasts. This can be seen in the image below.
Then, once you've been picked up, if your driver isn't going to Taitung, communicate to them that you want to be dropped off at the location below, in Fenggang (楓港， Feng1gang3)。 In fact, it might be easiest to say that you want to be dropped off at this CPC petrol station (you can show them via google maps link here), which is across the road from the hitching spot.
The map below displays where to stand to hitch from Fenggang to Taitung. Note, if you're coming from Kenting make sure you DON'T stand on the entry ramp to the highway on the south side of Fenggang. There are barely any cars there!
Here's a pic of what it looks like from this hitching spot (stand on the 'X'). All cars going to Taitung will go onto the ascending ramp.
Ollie got picked up by literally the first passing car from here which took him all the way to Taitung.