I’m sure others will reply soon to help you out with your connectivity problem. It will help if you give a little more detail - confirm which version of the Bitcoin Gold wallet you’re running (or when you downloaded it), what kind of computer (PC, Mac, Linux), confirm that you made sure your computer’s firewall is open so that the wallet can connect to other nodes, etc.
My Bitcoin Gold Core wallet shows that it’s currently connecting to the network just fine when I hover over the network connection icon in the lower-right corner (circled in green, here):
That was a curious choice to make, sending coin to a wallet address before the wallet software was fully up and running - but, regardless, the coin transaction happens on the blockchain, not in the wallet software. The wallet software just shows you the status on the chain. The coin you sent should be fine, as long as you sent it to a valid address.
You can look at your actual wallet balance on the chain in real-time by entering your public wallet address in the blockchain explorer, here. (Never share your private key with anyone.)