First, there is no SQL Server 2024. The current in-market version is SQL Server 2022. SQL Server 2025 is in preview, expected to be release before the year ends.
The best approach to move the database is by BACKUP/RESTORE. Testing is essential. The lowest compatibility level supported is 100, that is the level of SQL 2008. Furthermore, some older features have been dropped entirely, so if you are using any of them, you will have some work to do.
I would recommend that you aim for compatibility level 160, that is the most recent. This can have some repercussions for performance, which you may or may not find during testing. But you should test with compat level 160. You should also enable Query Store the first thing you do.
Once you have completed testing, you can start thinking on how to do the actual cutover. But one tip for performance: Again, enable Query Store first thing. But stay at compat level 100 for two weeks. Then switch to compat level 160. If you get performance regressions, you can force the old plan with help of Query Store to solve the urgent problem, while you are working on a long-time solution.