So when I was playing around with my wallets Friday night, I tested sending $1 from one to another. I opted not to pay a fee, figuring that I was just testing transactions and didn't care how long it took. However, the transaction is still listed on blockchain as unverified with unknown network propagation. Does anyone have a good estimate of how long the transaction will take to clear or a way to speed it up? Would another larger transaction going to the same wallet bring this small transaction with it? Not a huge deal, but it's just odd.
Which wallet did you use that allowed you to send the transaction without a fee?
Once the transaction is sent, there is currently no easy way to "speed it up". It could sit there for hours, days, or even weeks. In theory it could sit there unconfirmed forever.
A completely separate transaction that just happens to be going to (or coming from) the same address will have no effect at all on the likelihood of that previous transaction being confirmed.
Hopefully in the future miners will look not only at the fees on a transaction, but also at the fees on transactions that use the previous transaction as an input. This would allow recipients to create a fee that would expedite transactions when the sender fails to do so. Unfortunately, as far as I know, there are no miners that do this currently.
As has been mentioned already, depending on the wallet you used to create the transaction, it is possible that your wallet will eventually stop trying to broadcast the transaction and it will slowly disappear from the memory pool of all the peers on the network. At that time the bitcoins will revery back to the wallet that you attempted to send them from and the transaction will no longer exist.
If you are not using a wallet with this feature, then it is technically possible to replace the previous transaction with a new transaction that uses at least one of the same inputs but which includes a fee. It isn't an easy process, essentially you are attempting a double-spend attack on yourself, but it can be done.