The NET Bible notes:
Beth Aven means “house of wickedness” in Hebrew; it is a polemic reference to “Bethel,” which means “house of God.” Cf. CEV “at sinful Bethel.”
Bethel was also a worship center established by Jeroboam for the people of the Northern Kingdom. Hosea seems to be referring to the same place where a golden calf was worshiped, sacrifices where made to idols, and ritual prostitution was practiced.
Wikipedia suggests several referents to Gilgal:
- The location on the east side of the Jordan where Israel encamped mentioned in Joshua.
- A stop on Samuel's annual circuit mentioned in Samuel.
- The home of prophets associated with Elijah and Elisha.
It also suggests the Hebrew word means circle of standing stones, but according to Strongs, it simply means wheel. I don't know what to make of the difference or even if there is a difference of opinion.
By the time of Hosea, Amos, and Micah, the place (or at least place name) was clearly associated with sinfulness and especially illicit sacrifice. If altars were still being used at Gilgal after the temple was established in Jerusalem, then travelling there would represent the breaking of the covenant.
Both places are associated with idolatry and wickedness during the Divided Kingdom period. In the larger context of his prophetic work, Hosea illustrates Israel's separation from God by marrying a prostitute named Gomer who is unfaithful to him. The broken relationship also seems to symbolize the Northern Kingdom's separation from Judah, which is sometimes shown to be more faithful to God in the text (e.g., Hosea 11:12).