To buy new gear you will need to use the Market Wards. In Limsa you will find the ward entrance south of the aetheryte on the top floor. It is marked on the map as "Southern Islands".
Use the item search counter to find which wards/rows have the gear in that you are looking for and the price.

For a rank 10 Gladiator you will probably want the following gear or similar (although sadly lower rank gear doesn't tend to look pretty!):

Tarnished Hoplon (Rank 14)- Battlecraft Row
Dodoskin Jacket (Rank 15)- Armorfitters Row
Bronze Sallet (Rank 15)- Armorfitters Row
Dodoskin Jackboots (Rank 15)- Armorfitters Row
Sheepskin Armguards (Rank 15)- Armorfitters Row
Sheepskin Subligar (Rank 13)- Lower Tailors Row

All of the gear listed above favours your class (Gladiator).
I wouldn't equip any gear more than 5 ranks higher than your current rank, as you will not get the same benefit from it.
I hope that helps

Oh, on a side note, Limsa and Gridania wards tend to be pretty quiet so you will probably be best off heading to Uld'ah where the wards are used more.