Dolomite has a competitive fee structure, charging 0.5% for takers and -0.1% for makers (makers earn 0.1%). There are some additional small flat fees that may be applied to your trades which are outlined below, all payable in Ether (ETH).
Before each transaction Dolomite wraps your Ether such that it can be used to make a trade. Because of the design of Ether, it can't be used as-is to perform trades on a decentralized exchange, so it needs to be wrapped first. This wrapping costs a tiny flat fee to perform. Advanced traders can use the advanced menu on the Dolomite Exchange to pre-wrap a large amount of Ether so it doesn't need to be wrapped for each transaction if they wish. You can read more about wrapped ETH here: https://weth.io/.
Before tokens can be traded on a decentralized exchange such as Dolomite, the wallet first needs to "allow" them to be traded. This is a one-time transaction that will take place the first time you trade with a token. It is a tiny flat fee, and once performed will not need to be done again.
Transactions on the Ethereum blockchain take a "gas" fee in Ether to execute. This fee changes slightly over time depending on the gas price, but is generally a small fee of less than 1USD. Any trade done on a decentralized exchange such as Dolomite will require this fee to execute, and sending any tokens or Ether from your wallet will also require this fee.