Blockchain.info says that right now (1/14/2018) there are approximately 42,000 transactions awaiting confirmation. It also says that the average size of a block is 1,700. If the target is 10 minutes per block, that suggests that there are roughly 4 hours of transactions yet to be confirmed. If a transaction carries with it an "average fee", is it reasonable to assume that it will take approximately 4 hours for the first confirmation, and an hour more for 6 confirmations, given these numbers? Is there any reasonable way to estimate how that time would change given a change in the fee (other than just accepting some website's recommended fee)?
This is an impossible question to answer, because there will likely be new transactions added to the mempool continuously.
If more transactions get confirmed than are added, the mempool will shrink in size, and lower fee txs will confirm.
If less transactions get confirmed than are added, the mempool will grow in size, and it is likely that higher fee txs will confirm.
As a user, the total number of transactions shouldn't be of concern to you, instead you should look at what the fee rates of currently confirming transactions are, and plan accordingly. If your tx needs to be confirmed ASAP, you should set a fee slightly higher than what you are currently seeing. If you can afford to wait, you can set the fee lower, and hope that the mempool will shrink in size (but this is fundamentally unpredictable, it depends on the actions of all other useres).
Ultimately, you'll really only care about the number of txs paying a higher rate than you are willing to. Those txs are the ones that will affect how long your tx takes to confirm.