The Dolomites are vast, so it’s a bit difficult to provide recommendations that will truly match your interests without knowing more about the types of hikes you’d like to do and the sights you want to see. That said, I’d suggest basing yourself in either Val Pusteria or Val Gardena.
In Val Pusteria, consider staying near Toblach, San Candido, or Sexten. From there, you can hike iconic destinations such as Tre Cime di Lavaredo, and explore alpine lakes like Lago di Dobbiacco and Lago di Landro. If you're looking for a more laid-back hike, visit Val Fiscalina. It's a gorgeous valley that leads into the Sexten Dolomites. You can even hike through it to Tre Cime.
In Val Gardena, look into staying near Ortisei, S. Cristina, Selva, or Lajen. From these towns, you can access spectacular hikes such as Seceda, Alpe di Siusi, and Sassolungo. The Sassolungo circuit is a challenging hike you might enjoy.
Since you're visiting during the peak season, and if you’d like to avoid crowds, consider staying in the heart of Alta Badia. It's more of a hidden gem imo. La Val is a secluded mountain village that offers numerous hikes right outside your door. We loved our time there! This article will give you a good sense of the area: https://throneandvine.com/la-val-alta-badia/. Happy planning!