Looking for a service that can email/sms you if a monitored bitcoin/altcoin address has any moved funds. Is there such a service for this?
Check out blockcypher.com's API, they provide webhook callbacks (i.e. HTTPS notifications) for different types of events, including transactions being confirmed or attaining a certain confidence level.
I think block.io's API can do something similar, however that is more oriented around their own web wallet, i.e. for monitoring addresses in a wallet managed through them, rather than arbitrary addresses.
bitnotify.com and even blockchain.info offer such services. Note, however, that you should still go check whether you actually have received the money (which you can also do on blockchain.info) because the fact that you received a notification only means that the transaction was broadcasted, not that it made it into the blockchain, especially not that it made it into the blockchain permanently.