Today iwas my free day, we go. We went with my mum to repair our bikes.

When we arrived there they told us they couldn't repair our bikes because they didn't have a wheel.

After that, we went to eat pizza in theat a restaurant.

My free dayday off

Especially if you normally would have gone to work today, it is more common to say "day off" instead of "free day."

Today is my free day, we goday off, so we went with my mum to repair our bikes.

When we arrived there, they told us they couldn't repair our bikes because they didn't have a wheel.

"There" sounds better because, presumably, you are talking about your experience from somewhere other than the bike shop.

"... they didn't have a wheel" is grammatically correct but sounds strange. It sounds like they may have had wheels, but needed one more to complete the repair, or that they were lacking a specific kind of wheel for your bikes. If they were out of wheels, you could just say "they didn't have any wheels."

We went to another repair center and finally repairehad my mum's bike repaired.

Nothing technically wrong with what you wrote, but using the active voice without a subject makes it sound like you guys repaired your mom's bike yourselves instead of having the bike repair shop do it.

After that, we went to eat pizza in theat a restaurant.

Today is my free day, we goand we went with my mum to repair our bikes.

When we arrived here, they told us they couldn't repair our bikes because they didn't have a wheel.

After that, we went to eat pizza in theat a restaurant.