If I want to check if my transaction has made it to the blockchain, where can I look it up and by what data can I search/filter? Of course there is the transaction id and possibly a payment id, but is there other data I could use to find my transaction?
The best general tool for this is http://moneroblocks.info/. It is somewhat comparable to the Bitcoin site https://blockchain.info/, and you can conduct searches using a number of different identifiers.
However, if you are focusing on a specific transaction, I think your best bet might actually be a very useful tool developed by one of our community members: http://xmrtests.llcoins.net/checktx.html
By using the transaction hash, viewkey or private transaction key, and the recipient's public address you can prove that a transaction made it to its intended destination.