You cannot deposit USD, EUR, etc directly into Coinomi. You can buy crypto currency with fiat currency directly from Coinomi via third party services. To buy coins, simply open the left side menu and select "Buy (Credit Card)". Coinomi doesn't hold any fiat currency. All assets controlled by Coinomi are crypto assets, that may or may not have a monetary value.
While it's not exactly FIAT currency, you can buy gift cards to hundreds of different stores and services by selecting "Gift Cards". To withdraw any cryptocurrency to your bank account, you should first open an account in a crypto-exchange that works with your bank, and transfer your coins to be sold there. Another option is to trade with cryptocurrencies directly, by exchanging them for goods or services with merchants that accept them.
If you already have a supported asset in Coinomi, you can exchange it to another. Simply open the left-hand side menu and select one of our listed integrated exchanges. If you don't find the asset you want to trade on the exchanges' lists inside the app, please read here.
Good old fashioned work
Of course you can be part of the revolution and start trading crypto for goods and services directly. You can ask your employer to start receiving payments in crypto, and suggest paying in crypto with people/companies you usually purchase things from.