So with the Arcane Compass used, just follow the arrows, and they will eventually lead you to the portal. If it leads to a random spot on an island where there is nothing, chances are that it is underground. You can either go look for the cave entrance yourself, or just dig a straight path down and hope you don't eventually fall to your death in the portal room below you.
How to get an Arcane CompassThe Arcane Compass can be crafted from the Workbench lvl 2. It requires pretty basic crafting materials, and the full list of crafting materials are as following:
1x Copper Bar. Create this at the Furnace with 3x Copper Ore and 1x Coal.
1x Verdant Emerald. Find this by mining on low-level islands.
8x Blue Portal Shards. Find these by killing monsters in low-level islands.
Now craft the Arcane Compass and put it on your action bar. Use it at any point to spawn the arrows that will points towards the closest unopened portal on an island.