Thank you again everyone for your responses.
I am just trying to understand.
1) So I convert all my altcoins to one of the coins that Coinbase supports. I already have a Coinbase account. Can you tell me which one of the Coinbase supported coins has the lowest fees for this transaction and what would be the amount/percentage of the fee?
2) Can't I sell the coin (a supported coin) on Coinbase and then move the money from Coinbase to my bank?
You are correct on both.
Cheapest fees? I think it doesn't matter because Coinbase fees are the same percentage regardless of the coin. The only time it would matter for the mining fees. Make sure you check the withdrawal fee on bittrex. They are expensive for bitcoin, for example.
Have you managed to verify your account on bittrex? I can't verify and have 25% of all my wealth on there.