You will need to upgrade and evolve your broom to change its tier and stats, from the character menu > Broom section.
You will receive a Tier 1 "Old Broom" early in-game by following the main quest line.
However you will need a Tier 2 Broom to fly freely and a Tier 3 or higher to access all the flying related content.
You can upgrade your broom to +10 by using Crystals (Soulbound or not).
Once your broom is +10, you can evolve it to the next tier with Shining Crystals (Soulbound or not).
If you use the "Auto Upgrade" or "Auto Evolution" option, the game will automatically consume all the Crystals in your possession so you don't have to tap the upgrade/evolve button multiple times.
The Upgrade or Evolution can't fail, however the higher the tier, the more materials you need!