Upgrading your bag space in ESO costs gold and with each tier bought the price increases drastically.
To upgrade your bag space you must
first find the merchant, each starting town has a merchant just look for this icon on the right.
Everyone starts off with 60 inventory slots for their bag and the maximum is 110. Below is a table with the prices for upgrades.
Stonefalls Merchant Location
Auridon Merchant Location
Glenumbra Merchant Location