What rewards come from spending stamps?

The milestones count stamps spent, so every purchase moves you toward them.

That makes the milestone rewards a bonus on top of what you were buying anyway. You get the Flame Plated Dragon Shield once you have spent enough. The 3,500 stamp milestone pays out an appearance pack, Prismana orbs and phenomena crystals together. Spending more after that unlocks more items.

The Prismana orbs are worth having. A Prismana Orb upgrades a standard Aniimo into its Prismana form. The shop's seasonal section also sells them, so the milestone isn't the only way to get one.

Reaching the milestones without wasting stamps

You earn stamps from the activities listed under Stamp Rush in the Institute Guide, such as turning in Aniimo reports and Ultra Aniimo battles. Going to the Starcross Tea Party at the Astra coffee shop also gives you Windborne Stamps passively.

The shop's stock resets every week, and anything you don't buy that week is lost. The best way to push your spent total toward the milestones is to buy the weekly items you would lose anyway. Vein Essence is the clearest example: the shop sells 10 a week, and it's the fastest source of it. That way the stamps count toward a milestone and get you something useful at the same time.

Not settled: The game doesn't say exactly how many stamps you have to spend to unlock the Flame Plated Dragon Shield.