You’re able to see your balances directly from the Home Page of your Slingshot Wallet. Here you can view your updated token balances.
The home page aggregates balances across networks. If you would like to view specific network balances, you can click on a token and scroll down, where your balance will show per network.
Each time you switch networks, you’ll be able to see the current balance of selected network before swapping, as shown below.