I generated an account using geth on my PC. I was never fully synced with the blockchain. Can I access my wallet from another client? How would that be?
If the account was generated using geth. It's stored in a .json file under the keystore directory. The .json file contain the password encrypted private key. You can take the .json file and move it to a new client freely (into a new keystore directory in the new computer) or to use on online wallets.
I would recommand you to use Metamask https://metamask.io/ It is easy to use and configurable.
By default, it will connect you through some nodes. But you can also specify which node to connect to.