Hello! We didn't open our Roth IRA until January 2023. Do we report our contributions on our taxes this year, or will those go on next year?

You can choose to designate it as the 2022 contribution if you want to make additional contributions for tax year 2023.

You don't get to choose after the fact (unless by April 18, 2023 you ask the IRA custodian to change the year for which you made the contribution).  The contribution goes on your 2022 tax return only if you told the IRA custodian that the contribution was to be for 2022 and the IRA custodian recorded it as such, otherwise it's a 2023 contribution that goes on your 2023 tax return.