In battle I was just wondering how you raise the tier of your pet. Can anybody help? Please and thanks!

Tiers are based on your pets' stats, so you increase their stats and eventually you'll raise their tier. You can increase their stats by training them at the Training Center.

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Here's a thread that shows the ranges for the different tiers. You might find this helpful for knowing how far you are from the next tier you're working on :)

Thank you!