It’s a big step moving out of your parents’ house, so congrats! If you get a renters insurance policy, it will cover collectibles.
That said, many renters and home insurance policies have special limits of liability for money and coins. Typically, there is a $200 limit, meaning if something were to happen, your insurance would pay out $200. If your collection is worth more than this, you may want to avoid bringing it with you.
You should check with multiple insurance companies to determine who offers the highest limits for collectibles. You can do this by requesting quotes online to find the best rate for the coverage you need.