Learn Swedish: Vi gav oss ut på isen trots varningen. - We set out onto the ice despite the warning.

This is part of the reflexive phrasal verb att ge sig ut, which means to set out, venture out, or head out. Literally, it translates to "to give oneself out." Since the subject is vi (we), the reflexive pronoun becomes oss, and the past tense of ge is gav.

Could I use trots att instead of just trots in this sentence?

No, because trots att is a conjunction that must be followed by a full clause (a subject and a verb), like "despite the fact that...". Since varningen is just a noun, you must use the preposition trots (despite) on its own.

Does på mean "on" or "onto" here?

It translates best to "onto" in this context because the verb phrase gav oss ut implies motion toward a destination. Swedish often uses the same preposition (på) for both the static position ("on the ice") and the movement toward it ("onto the ice").

How does ge sig ut compare to other verbs like ge upp?

While ge upp means to give up, adding the reflexive pronoun changes the meaning entirely. Ge sig alone can mean to surrender or subside, ge sig av means to leave or depart, and ge sig ut, as seen here, means to set out.